CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Lecturer: Pavel Rytir The City College of New York 2015 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Cartesian Products and Relations Cartesian Product Let A and B be sets. The Cartesian product, or cross product, of A and B is denoted by A × B and is defined by {(x, y )|x ∈ A, y ∈ B}. Relation For sets A and B, a (binary) relation from A to B is a subset of A × B. Theorem 5.1 For any sets A, B, C ⊆ U. A × (B ∩ C ) = (A × B) ∩ (A × C ) A × (B ∪ C ) = (A × B) ∪ (A × C ) (A ∩ B) × C ) = (A × C ) ∩ (B × C ) (A ∪ B) × C ) = (A × C ) ∪ (B × C ) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 1 If A = {1, 2, 3, 4}, B = {2, 5}, and C = {3, 4, 7}, determine A × B; B × A; A ∪ (B × C ); (A ∪ B) × C ; (A × C ) ∪ (B × C ). Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 1 If A = {1, 2, 3, 4}, B = {2, 5}, and C = {3, 4, 7}, determine A × B; B × A; A ∪ (B × C ); (A ∪ B) × C ; (A × C ) ∪ (B × C ). Answer A × B = {(1, 2), (2, 2), (3, 2), (4, 2), (1, 5), (2, 5), (3, 5), (4, 5)}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 1 If A = {1, 2, 3, 4}, B = {2, 5}, and C = {3, 4, 7}, determine A × B; B × A; A ∪ (B × C ); (A ∪ B) × C ; (A × C ) ∪ (B × C ). Answer A × B = {(1, 2), (2, 2), (3, 2), (4, 2), (1, 5), (2, 5), (3, 5), (4, 5)}. B × A = {(2, 1), (2, 2), (2, 3), (2, 4), (5, 1), (5, 2), (5, 3), (5, 4)}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 1 If A = {1, 2, 3, 4}, B = {2, 5}, and C = {3, 4, 7}, determine A × B; B × A; A ∪ (B × C ); (A ∪ B) × C ; (A × C ) ∪ (B × C ). Answer A × B = {(1, 2), (2, 2), (3, 2), (4, 2), (1, 5), (2, 5), (3, 5), (4, 5)}. B × A = {(2, 1), (2, 2), (2, 3), (2, 4), (5, 1), (5, 2), (5, 3), (5, 4)}. A ∪ (B × C ) = {1, 2, 3, 4, (2, 3), (2, 4), (2, 7), (5, 3), (5, 4), (5, 7)} CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 1 If A = {1, 2, 3, 4}, B = {2, 5}, and C = {3, 4, 7}, determine A × B; B × A; A ∪ (B × C ); (A ∪ B) × C ; (A × C ) ∪ (B × C ). Answer A × B = {(1, 2), (2, 2), (3, 2), (4, 2), (1, 5), (2, 5), (3, 5), (4, 5)}. B × A = {(2, 1), (2, 2), (2, 3), (2, 4), (5, 1), (5, 2), (5, 3), (5, 4)}. A ∪ (B × C ) = {1, 2, 3, 4, (2, 3), (2, 4), (2, 7), (5, 3), (5, 4), (5, 7)} (A ∪ B) × C ) = (A × C ) ∪ (B × C ) = {(1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (1, 7), (2, 7), (3, 7), (4, 7), (5, 7)} CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 3 Let A = {1, 2, 3, 4}, and let B = {2, 4, 5}. Determine the following (a) |A × B| (b) the number of relations from A to B. (c) the number of relations on A. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 3 Let A = {1, 2, 3, 4}, and let B = {2, 4, 5}. Determine the following (a) |A × B| (b) the number of relations from A to B. (c) the number of relations on A. Answer (a) |A × B| = |A||B| = 12 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 3 Let A = {1, 2, 3, 4}, and let B = {2, 4, 5}. Determine the following (a) |A × B| (b) the number of relations from A to B. (c) the number of relations on A. Answer (a) |A × B| = |A||B| = 12 (b) The relation from A to B is a subset of A × B, there are 212 relations from A to B. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 3 Let A = {1, 2, 3, 4}, and let B = {2, 4, 5}. Determine the following (a) |A × B| (b) the number of relations from A to B. (c) the number of relations on A. Answer (a) |A × B| = |A||B| = 12 (b) The relation from A to B is a subset of A × B, there are 212 relations from A to B. (c) Since |A × A| = 16, there are 216 relations on A CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 3 Continue Let A = {1, 2, 3, 4}, and let B = {2, 4, 5}. Determine the following (d) the number of relations from A to B that contain (1, 2) and (1, 5). (e) the number of relations from A to B that contain exactly five ordered pairs. (f) the number of relations on A that contain at least 13 elements. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 3 Continue Let A = {1, 2, 3, 4}, and let B = {2, 4, 5}. Determine the following (d) the number of relations from A to B that contain (1, 2) and (1, 5). (e) the number of relations from A to B that contain exactly five ordered pairs. (f) the number of relations on A that contain at least 13 elements. Answer (d) For other ten ordered paris in A × B, there are two choices: include it in the relation or leave it out. Hence there are 210 relations from A to B that contain (1, 2) and (1, 5) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 3 Continue Let A = {1, 2, 3, 4}, and let B = {2, 4, 5}. Determine the following (d) the number of relations from A to B that contain (1, 2) and (1, 5). (e) the number of relations from A to B that contain exactly five ordered pairs. (f) the number of relations on A that contain at least 13 elements. Answer (d) For other ten ordered paris in A × B, there are two choices: include it in the relation or leave it out. Hence there are 210 relations from A to B that contain (1, 2) and (1, 5) (e) 12 5 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 3 Continue Let A = {1, 2, 3, 4}, and let B = {2, 4, 5}. Determine the following (d) the number of relations from A to B that contain (1, 2) and (1, 5). (e) the number of relations from A to B that contain exactly five ordered pairs. (f) the number of relations on A that contain at least 13 elements. Answer (d) For other ten ordered paris in A × B, there are two choices: include it in the relation or leave it out. Hence there are 210 relations from A to B that contain (1, 2) and (1, 5) (e) 12 5 16 16 16 (f) 16 13 + 14 + 15 + 16 . CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 10 A rumor is spread as follows. The originator calls two people. Each of these people phones three friends, each of whom in turn class five associates. If no one receives more than one class, and no one calls the originator, how many people now know the rumor? How many phone calls were made? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 10 A rumor is spread as follows. The originator calls two people. Each of these people phones three friends, each of whom in turn class five associates. If no one receives more than one class, and no one calls the originator, how many people now know the rumor? How many phone calls were made? Answer The number of people knowing the rumor is 1+2+2×3+2×3×5 = 39. Since each call lets one more people know the rumor, there are 39 − 1 = 38 calls made. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 12 Let A, B be the sets with |B| = 3. If there are 4096 relations from A to B, what is |A|. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.1 Cartesian Products and Relations Problem 12 Let A, B be the sets with |B| = 3. If there are 4096 relations from A to B, what is |A|. Answer Because 4096 = 212 , |A × B| = 12. 12 Therefore, |A| = |A×B| |B| = 3 = 4. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Functions: Plain and One-to-One Function For nonempty sets A and B, a function, or mapping, f from A to B, denoted by f : A → B, is a relation R from A to B in which every element of A appears exactly once as the first component of an ordered pair in the relation. ∀x ∈ A∃y ∈ B((x, y ) ∈ R ∧ ∀z ∈ B((x, z) ∈ R → (y = z))) We often write f (a) = b when (a, b) ∈ R. Here b is the image of a under f , and a is a preimage of b. Domain and Range For a function f : A → B, A is the domain of f and B is the codomain of f . The range of f is the set of all elements in B appearing as the second component in the ordered pairs of f , or {y |y ∈ B, (x, y ) ∈ R}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Functions: Plain and One-to-One One-on-one function A function f : A → B is called one-to-one, or injective, if each element of B appears at most once as the image of an element of A. Theorem 5.2 Suppose A1 , A2 are subsets of A. We define f (A1 ) = {b ∈ B|b = f (a), for some a ∈ A1 }. Then the following hold. f (A1 ∪ A2 ) = f (A1 ) ∪ f (A2 ) f (A1 ∩ A2 ) ⊆ f (A1 ) ∩ f (A2 ) f (A1 ∩ A2 ) = f (A1 ) ∩ f (A2 ) when f is one-to-one. Restriction and Extension Let A1 ⊆ A. Suppose f : A1 → B and g : A → B are two functions such that f (a) = g (a) for all a ∈ A1 . The function f is a restriction of g to A1 and g is a extension of f to A. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 1 Determine whether or not each of the following relations is a function. If a relation is a function, find its range (a) {(x, y )|x, y ∈ Z, y = x 2 + 7}, a relation from Z to Z. (b) {(x, y )|x, y ∈ R, y 2 = x}, a relation from R to R. (c) {(x, y )|x, y ∈ R, y = 3x + 1}, a relation from R to R. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 1 Determine whether or not each of the following relations is a function. If a relation is a function, find its range (a) {(x, y )|x, y ∈ Z, y = x 2 + 7}, a relation from Z to Z. (b) {(x, y )|x, y ∈ R, y 2 = x}, a relation from R to R. (c) {(x, y )|x, y ∈ R, y = 3x + 1}, a relation from R to R. Answer (a) Function. Range is {7, 8, 11, 16, 23, . . .} CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 1 Determine whether or not each of the following relations is a function. If a relation is a function, find its range (a) {(x, y )|x, y ∈ Z, y = x 2 + 7}, a relation from Z to Z. (b) {(x, y )|x, y ∈ R, y 2 = x}, a relation from R to R. (c) {(x, y )|x, y ∈ R, y = 3x + 1}, a relation from R to R. Answer (a) Function. Range is {7, 8, 11, 16, 23, . . .} (b) Not a function. For example, both (4, 2) and (4, −2) are in this relation. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 1 Determine whether or not each of the following relations is a function. If a relation is a function, find its range (a) {(x, y )|x, y ∈ Z, y = x 2 + 7}, a relation from Z to Z. (b) {(x, y )|x, y ∈ R, y 2 = x}, a relation from R to R. (c) {(x, y )|x, y ∈ R, y = 3x + 1}, a relation from R to R. Answer (a) Function. Range is {7, 8, 11, 16, 23, . . .} (b) Not a function. For example, both (4, 2) and (4, −2) are in this relation. (c) Function. Range is R. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 1 Continue Determine whether or not each of the following relations is a function. If a relation is a function, find its range (d) {(x, y )|x, y ∈ Q, x 2 + y 2 = 1}, a relation from Q to Q. (e) R is a relation from A to B where |A| = 5, |B| = 6, and |R| = 6. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 1 Continue Determine whether or not each of the following relations is a function. If a relation is a function, find its range (d) {(x, y )|x, y ∈ Q, x 2 + y 2 = 1}, a relation from Q to Q. (e) R is a relation from A to B where |A| = 5, |B| = 6, and |R| = 6. Answer (d) Not a function. For example, both (0, 1) and (0, −1) are in this relation. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 1 Continue Determine whether or not each of the following relations is a function. If a relation is a function, find its range (d) {(x, y )|x, y ∈ Q, x 2 + y 2 = 1}, a relation from Q to Q. (e) R is a relation from A to B where |A| = 5, |B| = 6, and |R| = 6. Answer (d) Not a function. For example, both (0, 1) and (0, −1) are in this relation. (e) Not a function, because |R| > |A|. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 2 Does the formula f (x) = 1/(x 2 − 2) define a function f : R → R? A function f : Z → R? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 2 Does the formula f (x) = 1/(x 2 − 2) define a function f : R → R? A function f : Z → R? Answer √ The √ formula cannot be used √ for the√domain of R since f ( 2) and f (− 2) are undefined. Since 2, − 2 ∈ / Z, the formula does define a real valued function on the domain Z. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Let A = {1, 2, 3, 4} and B = {x, y , z}. (a) List five functions from A to B. (b) How many functions f : A → B are there? (c) How many functions f : A → B are one-to-one? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Let A = {1, 2, 3, 4} and B = {x, y , z}. (a) List five functions from A to B. (b) How many functions f : A → B are there? (c) How many functions f : A → B are one-to-one? Answer (a) We list five functions as follows. {(1, x), (2, x), (3, x), (4, x)} {(1, x), (2, x), (3, x), (4, y )} {(1, x), (2, x), (3, x), (4, z)} {(1, x), (2, x), (3, y ), (4, x)} {(1, x), (2, x), (3, y ), (4, y )} CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Let A = {1, 2, 3, 4} and B = {x, y , z}. (a) List five functions from A to B. (b) How many functions f : A → B are there? (c) How many functions f : A → B are one-to-one? Answer (a) We list five functions as follows. {(1, x), (2, x), (3, x), (4, x)} {(1, x), (2, x), (3, x), (4, y )} {(1, x), (2, x), (3, x), (4, z)} {(1, x), (2, x), (3, y ), (4, x)} {(1, x), (2, x), (3, y ), (4, y )} (b) For each element in A, we can map it on an element in B. There are 34 functions from A to B. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Let A = {1, 2, 3, 4} and B = {x, y , z}. (a) List five functions from A to B. (b) How many functions f : A → B are there? (c) How many functions f : A → B are one-to-one? Answer (a) We list five functions as follows. {(1, x), (2, x), (3, x), (4, x)} {(1, x), (2, x), (3, x), (4, y )} {(1, x), (2, x), (3, x), (4, z)} {(1, x), (2, x), (3, y ), (4, x)} {(1, x), (2, x), (3, y ), (4, y )} (b) For each element in A, we can map it on an element in B. There are 34 functions from A to B. (c) Because |A| > |B|, there is no one-to-one function. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Continue Let A = {1, 2, 3, 4} and B = {x, y , z}. (d) How many functions g : B → A are there? (e) How many functions g : B → A are one-to-one? (f) How many functions f : A → B satisfy f (1) = x? (g) How many functions f : A → B satisfy f (1) = f (2) = x? (h) How many functions f : A → B satisfy f (1) = x and f (2) = y ? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Continue Let A = {1, 2, 3, 4} and B = {x, y , z}. (d) How many functions g : B → A are there? (e) How many functions g : B → A are one-to-one? (f) How many functions f : A → B satisfy f (1) = x? (g) How many functions f : A → B satisfy f (1) = f (2) = x? (h) How many functions f : A → B satisfy f (1) = x and f (2) = y ? Answer (d) 43 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Continue Let A = {1, 2, 3, 4} and B = {x, y , z}. (d) How many functions g : B → A are there? (e) How many functions g : B → A are one-to-one? (f) How many functions f : A → B satisfy f (1) = x? (g) How many functions f : A → B satisfy f (1) = f (2) = x? (h) How many functions f : A → B satisfy f (1) = x and f (2) = y ? Answer (d) 43 (e) P(4, 3) = 4! 1! = 24. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Continue Let A = {1, 2, 3, 4} and B = {x, y , z}. (d) How many functions g : B → A are there? (e) How many functions g : B → A are one-to-one? (f) How many functions f : A → B satisfy f (1) = x? (g) How many functions f : A → B satisfy f (1) = f (2) = x? (h) How many functions f : A → B satisfy f (1) = x and f (2) = y ? Answer (d) 43 (e) P(4, 3) = (f) 33 4! 1! = 24. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Continue Let A = {1, 2, 3, 4} and B = {x, y , z}. (d) How many functions g : B → A are there? (e) How many functions g : B → A are one-to-one? (f) How many functions f : A → B satisfy f (1) = x? (g) How many functions f : A → B satisfy f (1) = f (2) = x? (h) How many functions f : A → B satisfy f (1) = x and f (2) = y ? Answer (d) 43 (e) P(4, 3) = (f) 33 (g) 32 4! 1! = 24. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 3 Continue Let A = {1, 2, 3, 4} and B = {x, y , z}. (d) How many functions g : B → A are there? (e) How many functions g : B → A are one-to-one? (f) How many functions f : A → B satisfy f (1) = x? (g) How many functions f : A → B satisfy f (1) = f (2) = x? (h) How many functions f : A → B satisfy f (1) = x and f (2) = y ? Answer (d) 43 (e) P(4, 3) = (f) 33 (g) 32 (h) 32 4! 1! = 24. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 7 Determine each of the following. (a) b2.3 − 1.6c (b) b2.3c − b1.6c (c) d3.4eb6.2c (d) b3.4cd6.2e (e) b2πc (f) 2dπe Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 7 Determine each of the following. (a) b2.3 − 1.6c (b) b2.3c − b1.6c (c) d3.4eb6.2c (d) b3.4cd6.2e (e) b2πc (f) 2dπe Answer (a) 0 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 7 Determine each of the following. (a) b2.3 − 1.6c (b) b2.3c − b1.6c (c) d3.4eb6.2c (d) b3.4cd6.2e (e) b2πc (f) 2dπe Answer (a) 0 (b) 1 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 7 Determine each of the following. (a) b2.3 − 1.6c (b) b2.3c − b1.6c (c) d3.4eb6.2c (d) b3.4cd6.2e (e) b2πc (f) 2dπe Answer (a) 0 (b) 1 (c) 24 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 7 Determine each of the following. (a) b2.3 − 1.6c (b) b2.3c − b1.6c (c) d3.4eb6.2c (d) b3.4cd6.2e (e) b2πc (f) 2dπe Answer (a) 0 (b) 1 (c) 24 (d) 21 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 7 Determine each of the following. (a) b2.3 − 1.6c (b) b2.3c − b1.6c (c) d3.4eb6.2c (d) b3.4cd6.2e (e) b2πc (f) 2dπe Answer (a) 0 (b) 1 (c) 24 (d) 21 (e) 6 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 7 Determine each of the following. (a) b2.3 − 1.6c (b) b2.3c − b1.6c (c) d3.4eb6.2c (d) b3.4cd6.2e (e) b2πc (f) 2dπe Answer (a) 0 (b) 1 (c) 24 (d) 21 (e) 6 (f) 8 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 15 For each of the following functions, determine whether it is one-toone and determine its range. (a) f : Z → Z, f (x) = 2x + 1 (b) f : Q → Q, f (x) = 2x + 1 (c) f : Z → Z, f (x) = x 3 − x Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 15 For each of the following functions, determine whether it is one-toone and determine its range. (a) f : Z → Z, f (x) = 2x + 1 (b) f : Q → Q, f (x) = 2x + 1 (c) f : Z → Z, f (x) = x 3 − x Answer (a) One-to-one. The range is the set of all odd integers. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 15 For each of the following functions, determine whether it is one-toone and determine its range. (a) f : Z → Z, f (x) = 2x + 1 (b) f : Q → Q, f (x) = 2x + 1 (c) f : Z → Z, f (x) = x 3 − x Answer (a) One-to-one. The range is the set of all odd integers. (b) One-to-one. The range is Q. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 15 For each of the following functions, determine whether it is one-toone and determine its range. (a) f : Z → Z, f (x) = 2x + 1 (b) f : Q → Q, f (x) = 2x + 1 (c) f : Z → Z, f (x) = x 3 − x Answer (a) One-to-one. The range is the set of all odd integers. (b) One-to-one. The range is Q. (c) Since f (1) = f (0), it is not one-to-one. The range is {0, ±6, ±24, ±60, . . .} = {n3 − n|n ∈ Z}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 15 Continue For each of the following functions, determine whether it is one-toone and determine its range. (d) f : R → R, f (x) = e x (e) f : [−π/2, π/2] → R, f (x) = sin x (f) f : [0, π] → R, f (x) = sin x Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 15 Continue For each of the following functions, determine whether it is one-toone and determine its range. (d) f : R → R, f (x) = e x (e) f : [−π/2, π/2] → R, f (x) = sin x (f) f : [0, π] → R, f (x) = sin x Answer (d) One-to-one. The range is (0, +∞) = R+ . CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 15 Continue For each of the following functions, determine whether it is one-toone and determine its range. (d) f : R → R, f (x) = e x (e) f : [−π/2, π/2] → R, f (x) = sin x (f) f : [0, π] → R, f (x) = sin x Answer (d) One-to-one. The range is (0, +∞) = R+ . (e) One-to-one. The range is [−1, 1]. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 15 Continue For each of the following functions, determine whether it is one-toone and determine its range. (d) f : R → R, f (x) = e x (e) f : [−π/2, π/2] → R, f (x) = sin x (f) f : [0, π] → R, f (x) = sin x Answer (d) One-to-one. The range is (0, +∞) = R+ . (e) One-to-one. The range is [−1, 1]. (f) Since f (π/4) = f (3π/4), it is not one-to-one. The range is [0, 1]. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 17 Let A = {1, 2, 3, 4, 5}, B = {w , x, y , z}, A1 = {2, 3, 5} ⊆ A, and g : A1 → B. In how many ways can g be extended to a function f : A → B? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 17 Let A = {1, 2, 3, 4, 5}, B = {w , x, y , z}, A1 = {2, 3, 5} ⊆ A, and g : A1 → B. In how many ways can g be extended to a function f : A → B? Answer The extension must include f (1) and f (4). Since |B| = 4, there are 42 = 16 ways to extend the given function g . CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 20 If A = {1, 2, 3, 4, 5} and there are 6720 injective functions f : A → B, what is |B|? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 20 If A = {1, 2, 3, 4, 5} and there are 6720 injective functions f : A → B, what is |B|? Answer The number of injective (or one-to-one) functions from A to B is |B|! P(|B|, 5) = (|B|−5)! = 6720. Therefore, |B| = 8. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(a) One version of Ackermann’s function A(m, n) is defined recursively for m, n ∈ N by A(m, n) = n + 1 if m = 0, n ≥ 0 A(m, n) = A(m − 1, 1) if m > 0 and n = 0 A(m, n) = A(m − 1, A(m, n − 1)) if m > 0 and n > 0 Calculate A(1, 3) and A(2, 3). Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(a) One version of Ackermann’s function A(m, n) is defined recursively for m, n ∈ N by A(m, n) = n + 1 if m = 0, n ≥ 0 A(m, n) = A(m − 1, 1) if m > 0 and n = 0 A(m, n) = A(m − 1, A(m, n − 1)) if m > 0 and n > 0 Calculate A(1, 3) and A(2, 3). Answer A(1, 3) = A(0, A(1, 2)) = A(1, 2) + 1 = A(0, A(1, 1)) + 1 = A(1, 1) + 2 = A(0, A(1, 0)) + 2 = A(1, 0) + 3 = A(0, 1) + 3 = 2 + 3 = 5 Similarly, we have A(2, 3) = 9. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(b) One version of Ackermann’s function A(m, n) is defined recursively for m, n ∈ N by A(m, n) = n + 1 if m = 0, n ≥ 0 A(m, n) = A(m − 1, 1) if m > 0 and n = 0 A(m, n) = A(m − 1, A(m, n − 1)) if m > 0 and n > 0 Prove that A(1, n) = n + 2 for all n ∈ N. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(b) One version of Ackermann’s function A(m, n) is defined recursively for m, n ∈ N by A(m, n) = n + 1 if m = 0, n ≥ 0 A(m, n) = A(m − 1, 1) if m > 0 and n = 0 A(m, n) = A(m − 1, A(m, n − 1)) if m > 0 and n > 0 Prove that A(1, n) = n + 2 for all n ∈ N. Answer A(1, 0) = A(0, 1) = 2 = 0 + 2, the result holds for n = 0. Assume that A(1, k) = k + 2. Then A(1, k + 1) = A(0, A(1, k)) = A(0, k + 2) = k + 2 + 1 = (k + 1) + 2. Therefore, A(1, n) = n + 2 for all n ∈ N. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(c) One version of Ackermann’s function A(m, n) is defined recursively for m, n ∈ N by A(m, n) = n + 1 if m = 0, n ≥ 0 A(m, n) = A(m − 1, 1) if m > 0 and n = 0 A(m, n) = A(m − 1, A(m, n − 1)) if m > 0 and n > 0 For all n ∈ N show that A(2, n) = 3 + 2n. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(c) One version of Ackermann’s function A(m, n) is defined recursively for m, n ∈ N by A(m, n) = n + 1 if m = 0, n ≥ 0 A(m, n) = A(m − 1, 1) if m > 0 and n = 0 A(m, n) = A(m − 1, A(m, n − 1)) if m > 0 and n > 0 For all n ∈ N show that A(2, n) = 3 + 2n. Answer A(2, 0) = A(1, 1) = 1 + 2 = 3 = 3 + 2 × 0, the result holds for n = 0. Assume that A(2, k) = 3 + 2k. Then, A(2, k + 1) = A(1, A(2, k)) = A(2, k) + 2 = 3 + 2k + 2 = 3 + 2(k + 1). Therefore, A(2, n) = 3 + 2n for all n ∈ N. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(d) One version of Ackermann’s function A(m, n) is defined recursively for m, n ∈ N by A(m, n) = n + 1 if m = 0, n ≥ 0 A(m, n) = A(m − 1, 1) if m > 0 and n = 0 A(m, n) = A(m − 1, A(m, n − 1)) if m > 0 and n > 0 Verify that A(3, n) = 2n+3 − 3 for all n ∈ N. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(d) One version of Ackermann’s function A(m, n) is defined recursively for m, n ∈ N by A(m, n) = n + 1 if m = 0, n ≥ 0 A(m, n) = A(m − 1, 1) if m > 0 and n = 0 A(m, n) = A(m − 1, A(m, n − 1)) if m > 0 and n > 0 Verify that A(3, n) = 2n+3 − 3 for all n ∈ N. Answer A(3, 0) = A(2, 1) = 3 + 2 × 1 = 5 = 20+3 − 3, the result holds for n = 0. Assume that A(3, k) = 2k+3 −3. Then A(3, k +1) = A(2, A(3, k)) = 3+2A(3, k) = 3+2(2k−3 −3) = 2k−2 −3 = 2(k+1)−3 −3. Therefore, A(3, n) = 2n+3 − 3 for all n ∈ N. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(d) Given sets A, B, we define a partial function f with domain A and codomain B as a function from A0 to B where ∅¬A0 ⊂ A. (a) For A = {1, 2, 3, 4, 5}, B = {w , x, y , z}, how many partial functions have domain A and codomain B? (b) Let A, B be sets where |A| = m > 0, and |B| = n > 0. How many partial functions have domain A and codomain B? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(d) Given sets A, B, we define a partial function f with domain A and codomain B as a function from A0 to B where ∅¬A0 ⊂ A. (a) For A = {1, 2, 3, 4, 5}, B = {w , x, y , z}, how many partial functions have domain A and codomain B? (b) Let A, B be sets where |A| = m > 0, and |B| = n > 0. How many partial functions have domain A and codomain B? Answer (a) 54 44 + 5 3 43 + 5 2 42 + 5 1 41 = 55 − 45 − 1 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.2 Functions: Plain and One-to-One Problem 27(d) Given sets A, B, we define a partial function f with domain A and codomain B as a function from A0 to B where ∅¬A0 ⊂ A. (a) For A = {1, 2, 3, 4, 5}, B = {w , x, y , z}, how many partial functions have domain A and codomain B? (b) Let A, B be sets where |A| = m > 0, and |B| = n > 0. How many partial functions have domain A and codomain B? Answer (a) 54 44 + 53 43 + 52 42 + 51 41 = 55 − 45 − 1 m−1 m−2 m m (b) m−1 n + m−2 n + . . . + m1 n1 = (n + 1)m − nm − 1. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Onto Functions: Stirling Numbers of the Second Kind Onto function A function f : A → B is called onto, or surjective, if f (A) = B, that is, ∀y ∈ B∃x ∈ A(f (x) = y ). Stirling Number Let |A| = m and |B| = n where m ≥ b. The number of onto function from A to B is n! × S(m, n), in which S(m, n) is called a Stirling number of the second kind, and is defined by n n 1 X k ((−1) S(m, n) = (n − k)m n! n−k k=1 Theorem 5.3 Let m, n ∈ Z+ with 1 < n ≤ m. S(m + 1, n) = S(m, n − 1) + nS(m, n) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 1 Give an example of finite stes A and B with |A|, |B| ≥ 4 and a function f : A → B such that (a) f is neither one-to-one nor onto. (b) f is one-to-one and not onto. (c) f is onto but not one-to-one. (d) f is onto and one-to-one. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 1 Give an example of finite stes A and B with |A|, |B| ≥ 4 and a function f : A → B such that (a) f is neither one-to-one nor onto. (b) f is one-to-one and not onto. (c) f is onto but not one-to-one. (d) f is onto and one-to-one. Answer (a) A = {1, 2, 3, 4}, B = {a, b, c, d, e}, f = {(1, a), (2, a), (3, b), (4, c)} CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 1 Give an example of finite stes A and B with |A|, |B| ≥ 4 and a function f : A → B such that (a) f is neither one-to-one nor onto. (b) f is one-to-one and not onto. (c) f is onto but not one-to-one. (d) f is onto and one-to-one. Answer (a) A = {1, 2, 3, 4}, B = {a, b, c, d, e}, f = {(1, a), (2, a), (3, b), (4, c)} (b) A = {1, 2, 3, 4}, B = {a, b, c, d, e}, f = {(1, a), (2, b), (3, c), (4, d)} CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 1 Give an example of finite stes A and B with |A|, |B| ≥ 4 and a function f : A → B such that (a) f is neither one-to-one nor onto. (b) f is one-to-one and not onto. (c) f is onto but not one-to-one. (d) f is onto and one-to-one. Answer (a) A = {1, 2, 3, 4}, B = {a, b, c, d, e}, f = {(1, a), (2, a), (3, b), (4, c)} (b) A = {1, 2, 3, 4}, B = {a, b, c, d, e}, f = {(1, a), (2, b), (3, c), (4, d)} (c) A = {1, 2, 3, 4, 5}, B = {a, b, c, d}, f = {(1, a), (2, a), (3, b), (4, c), (5, d)} CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 1 Give an example of finite stes A and B with |A|, |B| ≥ 4 and a function f : A → B such that (a) f is neither one-to-one nor onto. Answer (a) A = {1, 2, 3, 4}, B = {a, b, c, d, e}, f = {(1, a), (2, a), (3, b), (4, c)} (b) A = {1, 2, 3, 4}, B = {a, b, c, d, e}, f = {(1, a), (2, b), (3, c), (4, d)} (b) f is one-to-one and not onto. (c) A = {1, 2, 3, 4, 5}, B = {a, b, c, d}, f = {(1, a), (2, a), (3, b), (4, c), (5, d)} (c) f is onto but not one-to-one. (d) A = {1, 2, 3, 4}, B = {a, b, c, d}, f = {(1, a), (2, b), (3, c), (4, d)} (d) f is onto and one-to-one. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 2 For each of the following functions f : Z → Z, determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range f (). (a) f (x) = x + 7. (b) f (x) = 2x − 3. (c) f (x) = −x + 5. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 2 For each of the following functions f : Z → Z, determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range f (). (a) f (x) = x + 7. (b) f (x) = 2x − 3. (c) f (x) = −x + 5. Answer (a) One-on-one and onto. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 2 For each of the following functions f : Z → Z, determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range f (). (a) f (x) = x + 7. (b) f (x) = 2x − 3. (c) f (x) = −x + 5. Answer (a) One-on-one and onto. (b) One-on-one but not onto. The range consists of all odd integers. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 2 For each of the following functions f : Z → Z, determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range f (). (a) f (x) = x + 7. (b) f (x) = 2x − 3. (c) f (x) = −x + 5. Answer (a) One-on-one and onto. (b) One-on-one but not onto. The range consists of all odd integers. (c) One-on-one and onto. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 2 Continue For each of the following functions f : Z → Z, determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range f (). (d) f (x) = x 2 . (e) f (x) = x 2 + x. (f) f (x) = x 3 . Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 2 Continue For each of the following functions f : Z → Z, determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range f (). (d) f (x) = x 2 . (e) f (x) = x 2 + x. (f) f (x) = x 3 . Answer (d) Not one-on-one since f (1) = f (−1). Not onto, the range is {0, 1, 4, 9, 16, . . .}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 2 Continue For each of the following functions f : Z → Z, determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range f (). (d) f (x) = x 2 . (e) f (x) = x 2 + x. (f) f (x) = x 3 . Answer (d) Not one-on-one since f (1) = f (−1). Not onto, the range is {0, 1, 4, 9, 16, . . .}. (e) Not one-on-one since f (0) = f (−1). Not onto, the range is {0, 2, 6, 12, 30, . . .}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 2 Continue For each of the following functions f : Z → Z, determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range f (). (d) f (x) = x 2 . (e) f (x) = x 2 + x. (f) f (x) = x 3 . Answer (d) Not one-on-one since f (1) = f (−1). Not onto, the range is {0, 1, 4, 9, 16, . . .}. (e) Not one-on-one since f (0) = f (−1). Not onto, the range is {0, 2, 6, 12, 30, . . .}. (f) One-to-one but not onto. The range is {. . . , −64, −27, −8, −1, 0, 1, 8, 27, 64, . . .}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 4 Let A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6}. (a) How many functions are there from A to B? How many of these are one-to-one? How many of these are onto? (b) How many functions are there from B to A? How many of these are one-to-one? How many of these are onto? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 4 Let A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6}. (a) How many functions are there from A to B? How many of these are one-to-one? How many of these are onto? (b) How many functions are there from B to A? How many of these are one-to-one? How many of these are onto? Answer (a) Each of element A can be mapped into one element in B, thus there are 64 functions from A to B. We need to select 4 elements in B corresponding to the elements in A with considering order, thus there are P(6, 4) = 6! 2! one-to-one functions. Since |B| > |A|, there is no onto functions. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 4 Let A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6}. (a) How many functions are there from A to B? How many of these are one-to-one? How many of these are onto? (b) How many functions are there from B to A? How many of these are one-to-one? How many of these are onto? Answer (a) Each of element A can be mapped into one element in B, thus there are 64 functions from A to B. We need to select 4 elements in B corresponding to the elements in A with considering order, thus there are P(6, 4) = 6! 2! one-to-one functions. Since |B| > |A|, there is no onto functions. (b) Similarly, there are 46 functions from B to A. Since |B| > |A|, there is no one-to-one function from B to A. There are 4! × S(6, 4) onto functions from B to A, where 1 Pn k n (n − k)m is the Stirling number. S(m, n) = n! (−1) k=0 n−k CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 9 Use the fact that every polynomial equation having real number coefficients and odd degree has a real root in order to show that the function f : R → R, defined by f (x) = x 5 − 2x 2 + x, is an onto function. Is f one to one? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 9 Use the fact that every polynomial equation having real number coefficients and odd degree has a real root in order to show that the function f : R → R, defined by f (x) = x 5 − 2x 2 + x, is an onto function. Is f one to one? Answer Consider the equation x 5 − 2x 2 + x − r = 0, where r is an arbitrary real number. Because this equation has real number coefficients and odd degree, it must have a real root. Thus, for any r ∈ R, there exists a real number (the root) to satisify that x 5 − 2x 2 + x = r , i.e., x 5 − 2x 2 + x is onto. However, since f (0) = f (1) = 0, f is not one-to-one. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 10 Suppose we have 7 different colored balls and 4 containers A, B, C and D. (a) In how many ways can we distributed the balls so that no container is left empty? (b) In this selection of 7 colored balls, one of them is blue. In how many ways can we distribute the balls so that no container is empty and the blue ball is in container B? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 10 Suppose we have 7 different colored balls and 4 containers A, B, C and D. (a) In how many ways can we distributed the balls so that no container is left empty? (b) In this selection of 7 colored balls, one of them is blue. In how many ways can we distribute the balls so that no container is empty and the blue ball is in container B? Answer (a) 4! × S(7, 4) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 10 Suppose we have 7 different colored balls and 4 containers A, B, C and D. (a) In how many ways can we distributed the balls so that no container is left empty? (b) In this selection of 7 colored balls, one of them is blue. In how many ways can we distribute the balls so that no container is empty and the blue ball is in container B? Answer (a) 4! × S(7, 4) (b) If B only contains one blue ball, there are 3! × S(6, 3) ways. If B contains more than one balls, there are 4! × S(6, 4) ways. Therefore, the answer is 3! × S(6, 3) + 4! × S(6, 4). CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 10 Continue Suppose we have 7 different colored balls and 4 containers A, B, C and D. (c) If we remove the letters on the containers so that we can no longer distinguish them, in how many ways can we distribute the 7 colored balls among the 4 identical containers, with some container(s) possibly empty? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 10 Continue Suppose we have 7 different colored balls and 4 containers A, B, C and D. (c) If we remove the letters on the containers so that we can no longer distinguish them, in how many ways can we distribute the 7 colored balls among the 4 identical containers, with some container(s) possibly empty? Answer (c) Suppose there are 0, 1, 2 and 3 containers not empty, the corresponding ways are S(7, 4), S(7, 3), S(7, 2) and S(7, 1) respectively. Therefore, the answer is S(7, 4) + S(7, 3) + S(7, 2) + S(7, 1). CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 15 A locker has n buttons labeled 1, 2, . . . , n. To open this lock we press each of n buttons exactly once. If no two or more buttons may be pressed simultaneously, then there are n! ways to do this. However, if one may press two or more buttons simultaneously, then there are more than n! ways to press all of the buttons. For instance, if n = 3, there are 3! = 6 ways to press the button one at a time, but there are 13 ways if we can press two or more buttons simultaneously. How many ways are there to press buttons when n = 4, n = 5, and more genereal case? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 15 A locker has n buttons labeled 1, 2, . . . , n. To open this lock we press each of n buttons exactly once. If no two or more buttons may be pressed simultaneously, then there are n! ways to do this. However, if one may press two or more buttons simultaneously, then there are more than n! ways to press all of the buttons. For instance, if n = 3, there are 3! = 6 ways to press the button one at a time, but there are 13 ways if we can press two or more buttons simultaneously. How many ways are there to press buttons when n = 4, n = 5, and more genereal case? Answer Imagine that each pressing action P corresponds to a container. Therefore, P when n = 4, we have 4i=1 i!S(4, i) ways; Pn when n = 5, we 5 have i=1 i!S(5, i) ways; in general, we have i=1 i!S(n, i) ways. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 15 Continue A locker has n buttons labeled 1, 2, . . . , n. To open this lock we press each of n buttons exactly once. If no two or more buttons may be pressed simultaneously, then there are n! ways to do this. However, if one may press two or more buttons simultaneously, then there are more than n! ways to press all of the buttons. For instance, if n = 3, there are 3! = 6 ways to press the button one at a time, but there are 13 ways if we can press two or more buttons simultaneously. If the lock has 15 buttons, and we need press 12 of them to open the lock, in how many ways can this be done? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.3 Onto Functions: Stirling Numbers of the Second Kind Problem 15 Continue A locker has n buttons labeled 1, 2, . . . , n. To open this lock we press each of n buttons exactly once. If no two or more buttons may be pressed simultaneously, then there are n! ways to do this. However, if one may press two or more buttons simultaneously, then there are more than n! ways to press all of the buttons. For instance, if n = 3, there are 3! = 6 ways to press the button one at a time, but there are 13 ways if we can press two or more buttons simultaneously. If the lock has 15 buttons, and we need press 12 of them to open the lock, in how many ways can this be done? Answer We choose 12 buttons from 15 buttons in P12 each of them, we have i=1 i!S(12, i) ways. P 12 15 i!S(12, i). i=1 12 15 12 ways. For The answer is CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 1 For A = {a, b, c}, let f : A × A → A be the closed binary operation given in the following table. f a b c a b a c b a c b c c b a Given an example to show that f is not associative. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 1 For A = {a, b, c}, let f : A × A → A be the closed binary operation given in the following table. f a b c a b a c b a c b c c b a Given an example to show that f is not associative. Answer f (f (a, b), c) = f (a, c) = c, while f (a, f (b, c)) = f (a, b) = a, thus, f is not associative. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 2 Let f : R × R → Z be the closed binary operation defined by f (a, b) = da + be. (a) Is f commutatitive? (b) Is f associative? (c) Does f have an identity element? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 2 Let f : R × R → Z be the closed binary operation defined by f (a, b) = da + be. (a) Is f commutatitive? (b) Is f associative? (c) Does f have an identity element? Answer (a) For all a, b ∈ R, f (a, b) = da + be = db + ae = f (b, a). Thus, f is commutative. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 2 Let f : R × R → Z be the closed binary operation defined by f (a, b) = da + be. (a) Is f commutatitive? (b) Is f associative? (c) Does f have an identity element? Answer (a) For all a, b ∈ R, f (a, b) = da + be = db + ae = f (b, a). Thus, f is commutative. (b) f (f (3.2, 4.7), 6.4) = f (8, 6.4) = 15, where f (3.2, f (4.7, 6.4)) = f (3.2, 12) = 16. Thus, f is not associative. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 2 Let f : R × R → Z be the closed binary operation defined by f (a, b) = da + be. (a) Is f commutatitive? (b) Is f associative? (c) Does f have an identity element? Answer (a) For all a, b ∈ R, f (a, b) = da + be = db + ae = f (b, a). Thus, f is commutative. (b) f (f (3.2, 4.7), 6.4) = f (8, 6.4) = 15, where f (3.2, f (4.7, 6.4)) = f (3.2, 12) = 16. Thus, f is not associative. (c) There is no identity element. Consider a = 0.5, then for any b ∈ R, f (a, b) ∈ Z. Suppose x is the identity element, we have 0.5 = f (0.5, x) ∈ Z, contradicting that 0.5 ∈ / Z. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 5 Let |A| = 5. (a) What is |A × A|? (b) How many functions f : A × A → A are there? (c) How many closed binary operations are there on A? (d) How many of these closed binary operations are commutative? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 5 Let |A| = 5. (a) What is |A × A|? (b) How many functions f : A × A → A are there? (c) How many closed binary operations are there on A? (d) How many of these closed binary operations are commutative? Answer (a) 5 × 5 = 25 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 5 Let |A| = 5. (a) What is |A × A|? (b) How many functions f : A × A → A are there? (c) How many closed binary operations are there on A? (d) How many of these closed binary operations are commutative? Answer (a) 5 × 5 = 25 (b) 525 . CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.4 Special Functions Problem 5 Let |A| = 5. (a) What is |A × A|? (b) How many functions f : A × A → A are there? (c) How many closed binary operations are there on A? (d) How many of these closed binary operations are commutative? Answer (a) 5 × 5 = 25 (b) 525 . (c) 525 , because every closed binary operation corresponds to a function of A × A → A. (d) Because there are 52 = 10 pairs of elements in A, there are 510 commutative closed binary operations. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 1 Larry returns from the laundromat with 12 pairs of socks (each pair a different color) in a laundry bag. Drawing the socks from the bag randomly, he will have to draw at most 13 of them to get a matched pair. What plays the roles of the pigeons and of the pigeonholes? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 1 Larry returns from the laundromat with 12 pairs of socks (each pair a different color) in a laundry bag. Drawing the socks from the bag randomly, he will have to draw at most 13 of them to get a matched pair. What plays the roles of the pigeons and of the pigeonholes? Answer The socks are the pigeons and the colors are the pigeonholes. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 2 Show that if 8 people are in a room, at least two of them have birthdays that occur on the same day of the week. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 2 Show that if 8 people are in a room, at least two of them have birthdays that occur on the same day of the week. Answer Follow the Pigeonhole Principle, where 8 people are pigeons and 7 days of a week are pigeonholes. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 3 An auditorium has a seating capacity of 800. How many seats must be occupied to guarantee that at least two people seated in the auditorium have the same first and the last initials? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 3 An auditorium has a seating capacity of 800. How many seats must be occupied to guarantee that at least two people seated in the auditorium have the same first and the last initials? Answer Because there are 262 = 676 pairs of the first and the last initials, the answer is 676 + 1 = 677. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 10 Let 4ABC be equilateral with AB = 1. Show that if we select 10 points in the interior of this triangle, there must be at least two points whose distance is less than 31 . Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 10 Let 4ABC be equilateral with AB = 1. Show that if we select 10 points in the interior of this triangle, there must be at least two points whose distance is less than 31 . Answer We can partition the 4ABC into 9 equilateral triangles with side length 13 . There must exist a small triangle containning two points, whose distance is definitely less than 13 . CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 11 Let ABCD be a square with AB = 1. Show that if we select five points in the interior of this square, there are at least two points whose distance is less than √12 . Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 11 Let ABCD be a square with AB = 1. Show that if we select five points in the interior of this square, there are at least two points whose distance is less than √12 . Answer We can partition the ABCD into 4 squares with side length 21 . There must exist a small square containning two points, whose distance is definitely less than √12 . CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 20 How many times must we roll a single die in order to get the same score (a) at least twice? (b) at least three times? (c) at least n times for n ≥ 4? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 20 How many times must we roll a single die in order to get the same score (a) at least twice? (b) at least three times? (c) at least n times for n ≥ 4? Answer (a) 7, to guarantee d7/6e = 2. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 20 How many times must we roll a single die in order to get the same score (a) at least twice? (b) at least three times? (c) at least n times for n ≥ 4? Answer (a) 7, to guarantee d7/6e = 2. (b) 13, to guarantee d13/6e = 3. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 20 How many times must we roll a single die in order to get the same score (a) at least twice? (b) at least three times? (c) at least n times for n ≥ 4? Answer (a) 7, to guarantee d7/6e = 2. (b) 13, to guarantee d13/6e = 3. (c) 6(n − 1) + 1, , to guarantee d 6(n−1)+1 e = n. 6 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 22 For m, n ∈ Z+ , prove that if m pigeons occupy n pigeonholes, then at least one pigeonhole has b m−1 n c + 1 or more pigeons roosting in it. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.5 The Pigeonhole Principle Problem 22 For m, n ∈ Z+ , prove that if m pigeons occupy n pigeonholes, then at least one pigeonhole has b m−1 n c + 1 or more pigeons roosting in it. Answer If not, each pigeonhole contains at most b(m − 1)/nc pigeons, for a total nb(m − 1)/nc ≤ m − 1 pigeons, contradicting that we have m pigeons. Therefore, the at least one pigeonhole has b m−1 n c + 1 or more pigeons roosting in it. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 1 (a) For A = {1, 2, 3 . . . , 7}, how many bijective functions f : A → A satisfy f (1) 6= 1? (b) Answer the part (a) where A = {x|x ∈ Z+ , 1 ≤ x ≤ n}, for some fixed n ∈ Z+ . Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 1 (a) For A = {1, 2, 3 . . . , 7}, how many bijective functions f : A → A satisfy f (1) 6= 1? (b) Answer the part (a) where A = {x|x ∈ Z+ , 1 ≤ x ≤ n}, for some fixed n ∈ Z+ . Answer (a) There are 7! bijective functions on A. Among them, there are 6! bijective funcitons satisfy f (1) = 1. Therefore, the answer is 7! − 6! = 6 × 6!. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 1 (a) For A = {1, 2, 3 . . . , 7}, how many bijective functions f : A → A satisfy f (1) 6= 1? (b) Answer the part (a) where A = {x|x ∈ Z+ , 1 ≤ x ≤ n}, for some fixed n ∈ Z+ . Answer (a) There are 7! bijective functions on A. Among them, there are 6! bijective funcitons satisfy f (1) = 1. Therefore, the answer is 7! − 6! = 6 × 6!. (b) In general, there are n! − (n − 1)! = (n − 1) × (n − 1)! bijective functions. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 4 Let g : N → N be defined by g (n) = 2n. If A = {1, 2, 3, 4} and f : A → N is given by f = {(1, 2), (2, 3), (3, 5), (4, 7)}, find g ◦ f . Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 4 Let g : N → N be defined by g (n) = 2n. If A = {1, 2, 3, 4} and f : A → N is given by f = {(1, 2), (2, 3), (3, 5), (4, 7)}, find g ◦ f . Answer g ◦ f = g (f (·)) = {(1, 4), (2, 6), (3, 10), (4, 14)}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 7(a) Let f , g , h : Z → Z be defined by f (x) = x − 1, g (x) = 3x and 0, x is even h(x) = 1, x is odd Determine f ◦ g , g ◦ f , g ◦ h, h ◦ g , f ◦ (g ◦ h), (f ◦ g ) ◦ h. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 7(a) Let f , g , h : Z → Z be defined by f (x) = x − 1, g (x) = 3x and 0, x is even h(x) = 1, x is odd Determine f ◦ g , g ◦ f , g ◦ h, h ◦ g , f ◦ (g ◦ h), (f ◦ g ) ◦ h. Answer (f ◦ g )(x) = 3x − 1; (g ◦ f )(x) = 3(x − 1); (g ◦ h)(x) = 0 when x is even, or 3 when x is odd. (h ◦ g )(x) = 0 when x is even, or 3 when x is odd. (f ◦ (g ◦ h))(x) = −1 when x is even, or 2 when x is odd. ((f ◦ g ) ◦ h)(x) = −1 when x is even, or 2 when x is odd. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 7(b) Let f , g , h : Z → Z be defined by f (x) = x − 1, g (x) = 3x and 0, x is even h(x) = 1, x is odd Determine f 2 , f 3 , g 2 , g 3 , h2 , h3 , h500 . Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 7(b) Let f , g , h : Z → Z be defined by f (x) = x − 1, g (x) = 3x and 0, x is even h(x) = 1, x is odd Determine f 2 , f 3 , g 2 , g 3 , h2 , h3 , h500 . Answer f 2 (x) = x − 2; f 3 (x) = x − 3; g 2 = 9x; g 3 = 27x; h2 (x) = h(x), h3 (x) = h(x), h500 (x) = h(x). CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 12 If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10, 12}, and f : A → B where f = {(1, 2), (2, 6), (3, 6), (4, 8), (5, 6), (6, 8), (7, 12)}, determine the primage of B1 under f in each of the following case. (a) B1 = {2}. (b) B1 = {6}. (c) B1 = {6, 8}. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 12 If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10, 12}, and f : A → B where f = {(1, 2), (2, 6), (3, 6), (4, 8), (5, 6), (6, 8), (7, 12)}, determine the primage of B1 under f in each of the following case. (a) B1 = {2}. (b) B1 = {6}. (c) B1 = {6, 8}. Answer (a) f −1 ({2}) = {a ∈ A|f (a) ∈ {2}} = {a ∈ A|f (a) = 2} = {1}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 12 If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10, 12}, and f : A → B where f = {(1, 2), (2, 6), (3, 6), (4, 8), (5, 6), (6, 8), (7, 12)}, determine the primage of B1 under f in each of the following case. (a) B1 = {2}. (b) B1 = {6}. (c) B1 = {6, 8}. Answer (a) f −1 ({2}) = {a ∈ A|f (a) ∈ {2}} = {a ∈ A|f (a) = 2} = {1}. (b) f −1 ({6}) = {2, 3, 5}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 12 If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10, 12}, and f : A → B where f = {(1, 2), (2, 6), (3, 6), (4, 8), (5, 6), (6, 8), (7, 12)}, determine the primage of B1 under f in each of the following case. (a) B1 = {2}. (b) B1 = {6}. (c) B1 = {6, 8}. Answer (a) f −1 ({2}) = {a ∈ A|f (a) ∈ {2}} = {a ∈ A|f (a) = 2} = {1}. (b) f −1 ({6}) = {2, 3, 5}. (c) f −1 ({6, 8}) = {2, 3, 4, 5, 6}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 12 Continue If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10, 12}, and f : A → B where f = {(1, 2), (2, 6), (3, 6), (4, 8), (5, 6), (6, 8), (7, 12)}, determine the primage of B1 under f in each of the following case. (d) B1 = {6, 8, 10}. (e) B1 = {6, 8, 10, 12}. (f) B1 = {10, 12}. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 12 Continue If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10, 12}, and f : A → B where f = {(1, 2), (2, 6), (3, 6), (4, 8), (5, 6), (6, 8), (7, 12)}, determine the primage of B1 under f in each of the following case. (d) B1 = {6, 8, 10}. (e) B1 = {6, 8, 10, 12}. (f) B1 = {10, 12}. Answer (d) f −1 ({6, 8, 10}) = {2, 3, 4, 5, 6}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 12 Continue If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10, 12}, and f : A → B where f = {(1, 2), (2, 6), (3, 6), (4, 8), (5, 6), (6, 8), (7, 12)}, determine the primage of B1 under f in each of the following case. (d) B1 = {6, 8, 10}. (e) B1 = {6, 8, 10, 12}. (f) B1 = {10, 12}. Answer (d) f −1 ({6, 8, 10}) = {2, 3, 4, 5, 6}. (e) f −1 ({6, 8, 10, 12}) = {2, 3, 4, 5, 6, 7}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 12 Continue If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10, 12}, and f : A → B where f = {(1, 2), (2, 6), (3, 6), (4, 8), (5, 6), (6, 8), (7, 12)}, determine the primage of B1 under f in each of the following case. (d) B1 = {6, 8, 10}. (e) B1 = {6, 8, 10, 12}. (f) B1 = {10, 12}. Answer (d) f −1 ({6, 8, 10}) = {2, 3, 4, 5, 6}. (e) f −1 ({6, 8, 10, 12}) = {2, 3, 4, 5, 6, 7}. (f) f −1 ({10, 12}) = {7}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 15 Let A = {1, 2, 3, 4, 5} and B = {6, 7, 8, 9, 10, 11, 12}. How many functions f : A → B are such that f −1 ({6, 7, 8}) = {1, 2}? Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 15 Let A = {1, 2, 3, 4, 5} and B = {6, 7, 8, 9, 10, 11, 12}. How many functions f : A → B are such that f −1 ({6, 7, 8}) = {1, 2}? Answer Since f −1 ({6, 7, 8}) = {1, 2}, there are three choices for each of f (1) and f (2), namely 6, 7, or 8. Furthermore, 3, 4, 5 ∈ / f −1 ({6, 7, 8}) so 3, 4, 5 ∈ f −1 ({9, 10, 11, 12}), and we have four choices for each of f (3), f (4) and f (5). Therefore, it follows by the rule of product that there are 32 ×43 = 576 funcitons from A to B where f −1 ({6, 7, 8}) = {1, 2}. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 20 (a) Given an example of a function f : Z → Z where f is one-to-one but not onto. (b) Given an example of a function f : Z → Z where f is onto but not one-to-one. (c) Do these examples contradict the Theorem 5.11, described as follows? Theorem 5.11 Let f : A → B for finite sets A and B where |A| = |B|. Then the following statements are equivalent: (i) f is one-to-one; (b) f is onto; (c) f is invertible. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 20 (a) Given an example of a function f : Z → Z where f is one-to-one but not onto. (b) Given an example of a function f : Z → Z where f is onto but not one-to-one. (c) Do these examples contradict the Theorem 5.11, described as follows? Theorem 5.11 Let f : A → B for finite sets A and B where |A| = |B|. Then the following statements are equivalent: (i) f is one-to-one; (b) f is onto; (c) f is invertible. Answer (a) f (x) = 2x. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 20 (a) Given an example of a function f : Z → Z where f is one-to-one but not onto. (b) Given an example of a function f : Z → Z where f is onto but not one-to-one. (c) Do these examples contradict the Theorem 5.11, described as follows? Theorem 5.11 Let f : A → B for finite sets A and B where |A| = |B|. Then the following statements are equivalent: (i) f is one-to-one; (b) f is onto; (c) f is invertible. Answer (a) f (x) = 2x. (b) f (x) = bx/2c. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 20 (a) Given an example of a function f : Z → Z where f is one-to-one but not onto. (b) Given an example of a function f : Z → Z where f is onto but not one-to-one. (c) Do these examples contradict the Theorem 5.11, described as follows? Theorem 5.11 Let f : A → B for finite sets A and B where |A| = |B|. Then the following statements are equivalent: (i) f is one-to-one; (b) f is onto; (c) f is invertible. Answer (a) f (x) = 2x. (b) f (x) = bx/2c. (c) No, Z is not a finite set. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 21 Let f : Z → N be defined by 2x − 1, if x > 0 f (x) = −2x, if x ≤ 0 (a) Prove that f is one-to-one and onto. Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 21 Let f : Z → N be defined by 2x − 1, if x > 0 f (x) = −2x, if x ≤ 0 (a) Prove that f is one-to-one and onto. Answer (a) When x ≥ 0, f (x) is odd. When x < 0, f (x) is even. Therefore, f is one-to-one. Let n ∈ N. If n is even, then (−n/2) < 0, (−n/2) ∈ Z, and f (−n/2) = n. If n is odd, then (n + 1)/2 > 0, (n + 1)/2 ∈ Z, and f ((n + 1)/2) = n. Therefore, f is onto. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 21 Continue Let f : Z → N be defined by 2x − 1, if x > 0 f (x) = −2x, if x ≤ 0 (b) Determine f −1 . Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.6 Function Composition and Inverse Functions Problem 21 Continue Let f : Z → N be defined by 2x − 1, if x > 0 f (x) = −2x, if x ≤ 0 (b) Determine f −1 . Answer (b) f −1 : N → Z is defined as (x + 1)/2 x is odd −1 f (x) = −x/2 x is even CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 1 Determine the best “big-Oh” form for each of the following functions f : Z+ → R. (a) f (n) = 3n + 7 (b) f (n) = 3 + sin(1/n) (c) f (n) = n3 − 5n2 + 25n − 165 (d) f (n) = 5n2 + 3n log2 n Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 1 Determine the best “big-Oh” form for each of the following functions f : Z+ → R. (a) f (n) = 3n + 7 (b) f (n) = 3 + sin(1/n) (c) f (n) = n3 − 5n2 + 25n − 165 (d) f (n) = 5n2 + 3n log2 n Answer (a) f ∈ O(n) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 1 Determine the best “big-Oh” form for each of the following functions f : Z+ → R. (a) f (n) = 3n + 7 (b) f (n) = 3 + sin(1/n) (c) f (n) = n3 − 5n2 + 25n − 165 (d) f (n) = 5n2 + 3n log2 n Answer (a) f ∈ O(n) (b) f ∈ O(1) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 1 Determine the best “big-Oh” form for each of the following functions f : Z+ → R. (a) f (n) = 3n + 7 (b) f (n) = 3 + sin(1/n) (c) f (n) = n3 − 5n2 + 25n − 165 (d) f (n) = 5n2 + 3n log2 n Answer (a) f ∈ O(n) (b) f ∈ O(1) (c) f ∈ O(n3 ) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 1 Determine the best “big-Oh” form for each of the following functions f : Z+ → R. (a) f (n) = 3n + 7 (b) f (n) = 3 + sin(1/n) (c) f (n) = n3 − 5n2 + 25n − 165 (d) f (n) = 5n2 + 3n log2 n Answer (a) f ∈ O(n) (b) f ∈ O(1) (c) f ∈ O(n3 ) (d) f ∈ O(n2 ) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 1 Continue Determine the best “big-Oh” form for each of the following functions f : Z+ → R. (e) f (n) = n2 + (n − 1)3 (f) f (n) = n(n+1)(n+2) (n+3) (g) f (n) = 2 + 4 + 6 + . . . + 2n Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 1 Continue Determine the best “big-Oh” form for each of the following functions f : Z+ → R. (e) f (n) = n2 + (n − 1)3 (f) f (n) = n(n+1)(n+2) (n+3) (g) f (n) = 2 + 4 + 6 + . . . + 2n Answer (e) f ∈ O(n3 ) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 1 Continue Determine the best “big-Oh” form for each of the following functions f : Z+ → R. (e) f (n) = n2 + (n − 1)3 (f) f (n) = n(n+1)(n+2) (n+3) (g) f (n) = 2 + 4 + 6 + . . . + 2n Answer (e) f ∈ O(n3 ) (f) f ∈ O(n2 ) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 1 Continue Determine the best “big-Oh” form for each of the following functions f : Z+ → R. (e) f (n) = n2 + (n − 1)3 (f) f (n) = n(n+1)(n+2) (n+3) (g) f (n) = 2 + 4 + 6 + . . . + 2n Answer (e) f ∈ O(n3 ) (f) f ∈ O(n2 ) (g) f ∈ O(n2 ) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 11 The following is analogous to the “big-Oh”. For f , g : Z+ → R we say that f is of order at least g if there exist constants M ∈ R+ and k ∈ Z+ such that |f (n)| ≥ M|g (n)| for all n ∈ Z+ where n ≥ k. In this case we write f ∈ Ω(g ) and says f is “big Omega of” g . So Ω(g ) represents the set of all functions with domain Z+ and codomain R that dominate g . Suppose that f , g , h : Z+ → R, where f (n) = 5n2 + 3n, g (n) = n2 , h(n) = n, for all n ∈ Z+ . Prove that (a) f ∈ Ω(g ) (b) g ∈ Ω(f ) Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 11 The following is analogous to the “big-Oh”. For f , g : Z+ → R we say that f is of order at least g if there exist constants M ∈ R+ and k ∈ Z+ such that |f (n)| ≥ M|g (n)| for all n ∈ Z+ where n ≥ k. In this case we write f ∈ Ω(g ) and says f is “big Omega of” g . So Ω(g ) represents the set of all functions with domain Z+ and codomain R that dominate g . Suppose that f , g , h : Z+ → R, where f (n) = 5n2 + 3n, g (n) = n2 , h(n) = n, for all n ∈ Z+ . Prove that (a) f ∈ Ω(g ) (b) g ∈ Ω(f ) Answer (a) Let M = 1 and k = 1. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 11 The following is analogous to the “big-Oh”. For f , g : Z+ → R we say that f is of order at least g if there exist constants M ∈ R+ and k ∈ Z+ such that |f (n)| ≥ M|g (n)| for all n ∈ Z+ where n ≥ k. In this case we write f ∈ Ω(g ) and says f is “big Omega of” g . So Ω(g ) represents the set of all functions with domain Z+ and codomain R that dominate g . Suppose that f , g , h : Z+ → R, where f (n) = 5n2 + 3n, g (n) = n2 , h(n) = n, for all n ∈ Z+ . Prove that (a) f ∈ Ω(g ) (b) g ∈ Ω(f ) Answer (a) Let M = 1 and k = 1. (b) Let M = 0.1 and k = 1. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 11 Continue The following is analogous to the “big-Oh”. For f , g : Z+ → R we say that f is of order at least g if there exist constants M ∈ R+ and k ∈ Z+ such that |f (n)| ≥ M|g (n)| for all n ∈ Z+ where n ≥ k. In this case we write f ∈ Ω(g ) and says f is “big Omega of” g . So Ω(g ) represents the set of all functions with domain Z+ and codomain R that dominate g . Suppose that f , g , h : Z+ → R, where f (n) = 5n2 + 3n, g (n) = n2 , h(n) = n, for all n ∈ Z+ . Prove that (c) f ∈ Ω(h) (d) h ∈ / Ω(f ) Answer CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 11 Continue The following is analogous to the “big-Oh”. For f , g : Z+ → R we say that f is of order at least g if there exist constants M ∈ R+ and k ∈ Z+ such that |f (n)| ≥ M|g (n)| for all n ∈ Z+ where n ≥ k. In this case we write f ∈ Ω(g ) and says f is “big Omega of” g . So Ω(g ) represents the set of all functions with domain Z+ and codomain R that dominate g . Suppose that f , g , h : Z+ → R, where f (n) = 5n2 + 3n, g (n) = n2 , h(n) = n, for all n ∈ Z+ . Prove that (c) f ∈ Ω(h) (d) h ∈ / Ω(f ) Answer (c) Let M = 1 and k = 1. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 11 Continue The following is analogous to the “big-Oh”. For f , g : Z+ → R we say that f is of order at least g if there exist constants M ∈ R+ and k ∈ Z+ such that |f (n)| ≥ M|g (n)| for all n ∈ Z+ where n ≥ k. In this case we write f ∈ Ω(g ) and says f is “big Omega of” g . So Ω(g ) represents the set of all functions with domain Z+ and codomain R that dominate g . Suppose that f , g , h : Z+ → R, where f (n) = 5n2 + 3n, g (n) = n2 , h(n) = n, for all n ∈ Z+ . Prove that (c) f ∈ Ω(h) (d) h ∈ / Ω(f ) Answer (c) Let M = 1 and k = 1. (d) Suppose h ∈ Ω(f ), i.e., there exists such M and k. Then we have 0 < M ≤ 5n2n+3n = 1, i.e., M cannot be a positive constant. Threfore, h ∈ / Ω(f ). CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 14 For f , g : Z+ → R, we say f is “big Theta of” g , and write f ∈ Θ(g ), when there exists constant m1 , m2 ∈ R+ and k ∈ Z+ such that m1 |g (n)| ≤ |f (n)| ≤ m2 |g (n)|, for all n ∈ Z+ where n ≥ k. Prove that f ∈ Θ(g ) if and only if f ∈ Ω(g ) and f ∈ O(g ). Answer When f ∈ Θ(g ), ∃m1 , m2 ∈ R+ ∃k ∈ Z+ ∀n ≥ km1 |g (n)| ≤ |f (n)| ≤ m2 |g (n)|. Therefore, ∃m1 ∈ R+ ∃k ∈ Z+ ∀n ≥ km1 |g (n)| ≤ |f (n)|, i.e., f ∈ Ω(g ). Similarly, we have f ∈ O(g ). Conversely, f ∈ Ω(g ) implies that ∃m1 ∈ R+ ∃k1 ∈ Z+ ∀n ≥ k1 m1 |g (n)| ≤ |f (n)|; f ∈ O(g ) implies that ∃m2 ∈ R+ ∃k2 ∈ Z+ ∀n ≥ k2 |f (n)| ≤ m2 |g (n)|. Let k = max{k1 , k2 }, we have ∃m1 , m2 ∈ R+ ∃k ∈ Z+ ∀n ≥ km1 |g (n)| ≤ |f (n)| ≤ m2 |g (n)|, i.e., f ∈ Θ(g ). CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 16 (a) Let f : Z+ → R where f (n) = (b) Let f : Z+ → R where g (n) = Pn Pi=1 n i. Prove that f ∈ Θ(n2 ). Prove that g ∈ Θ(n3 ). P (c) For t ∈ Z+ , let h : Z+ → R where h(n) = ni=1 i t . Prove that f ∈ Θ(nt+1 ). Answer i=1 i 2. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 16 (a) Let f : Z+ → R where f (n) = (b) Let f : Z+ → R where g (n) = Pn Pi=1 n i. Prove that f ∈ Θ(n2 ). Prove that g ∈ Θ(n3 ). P (c) For t ∈ Z+ , let h : Z+ → R where h(n) = ni=1 i t . Prove that f ∈ Θ(nt+1 ). i=1 i 2. Answer (c) f (n) = n(n+1) . 2 Let m1 = 0.2, m2 = 2 and k = 1. CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 16 (a) Let f : Z+ → R where f (n) = (b) Let f : Z+ → R where g (n) = Pn Pi=1 n i. Prove that f ∈ Θ(n2 ). Prove that g ∈ Θ(n3 ). P (c) For t ∈ Z+ , let h : Z+ → R where h(n) = ni=1 i t . Prove that f ∈ Θ(nt+1 ). i=1 i 2. Answer (c) f (n) = n(n+1) . Let m1 = 0.2, m2 = 2 and k = 1. P (d) f (n) ≤ i=1 n2 = n3 . f (n) ≥ ni=dn/2e i 2 ≥ Pn 2 2 3 i=dn/2e dn/2e = d(n + 1)/2edn/2e > n /8. Let k = 1, m1 = 1/8 and m2 = 1. Pn2 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.7 Computational Complexity Problem 16 (a) Let f : Z+ → R where f (n) = (b) Let f : Z+ → R where g (n) = Pn Pi=1 n i. Prove that f ∈ Θ(n2 ). Prove that g ∈ Θ(n3 ). P (c) For t ∈ Z+ , let h : Z+ → R where h(n) = ni=1 i t . Prove that f ∈ Θ(nt+1 ). i=1 i 2. Answer (c) f (n) = n(n+1) . Let m1 = 0.2, m2 = 2 and k = 1. P (d) f (n) ≤ i=1 n2 = n3 . f (n) ≥ ni=dn/2e i 2 ≥ Pn 2 2 3 i=dn/2e dn/2e = d(n + 1)/2edn/2e > n /8. Let k = 1, m1 = 1/8 and m2 = 1. P P (e) f (n) ≤ ni=1 nt = nt+1 . f (n) ≥ ni=dn/2e i t ≥ Pn t t t+1 . Let k = 1, i=dn/2e dn/2e = d(n + 1)/2edn/2e > (n/2) m1 = (1/2)t+1 and m2 = 1. Pn2 CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.8 Analysis of Algorithms Problem 1 (a) Define the time-complexity function f (n) to be the number of times the statement sum = sum + 1 is executed. Given the following program fragment, determine the best “big-Oh” form for f . sum=0; for i=1 to n do for j=1 to n do sum=sum+1; Answer f (n) = n2 , so f ∈ O(n2 ). CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.8 Analysis of Algorithms Problem 1 (b) Define the time-complexity function f (n) to be the number of times the statement sum = sum + 1 is executed. Given the following program fragment, determine the best “big-Oh” form for f . sum=0; for i=1 to n do for j=1 to n*n do sum=sum+1; Answer f (n) = n3 , so f ∈ O(n3 ). CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.8 Analysis of Algorithms Problem 1 (c) Define the time-complexity function f (n) to be the number of times the statement sum = sum + 1 is executed. Given the following program fragment, determine the best “big-Oh” form for f . sum=0; for i=1 to n do for j=i to n do sum=sum+1; Answer f (n) = n + (n − 1) + . . . + 1 = n(n+1) , 2 so f ∈ O(n2 ). CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.8 Analysis of Algorithms Problem 1 (d) Define the time-complexity function f (n) to be the number of times the statement sum = sum + 1 is executed. Given the following program fragment, determine the best “big-Oh” form for f . sum=0; i = n; while i¿0 do sum = sum + 1; i = bi/2c; Answer f ∈ O(log2 n) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.8 Analysis of Algorithms Problem 1 (e) Define the time-complexity function f (n) to be the number of times the statement sum = sum + 1 is executed. Given the following program fragment, determine the best “big-Oh” form for f . sum=0; for i=1 to n do j = n; while j ¿ 0 do sum = sum + 1; j = bj/2c; Answer f ∈ O(n log2 n) CSC 10400: Discrete Mathematical Structures Recitation Chapter 5 Chapter 5: Relations and Functions Section 5.8 Analysis of Algorithms Problem 5 The following pseudocode procedure can be used to evaluate the polynomial. product=1; value = a0 ; for i=1 to n do product=product*x; value=value+ai *product; For the polynomial 8−10x +7x 2 −2x 3 +3x 4 +12x 5 , we have n = 5, a0 = 8, a1 = −10, a2 = 7, a3 = −2, a4 = 3, and a5 = 12. (a) How many additions and how many multiplications occur to evaluate this polynomial? (b) How many additions and how many multiplications occur to evaluate the general polynomial a0 + a1 x + a2 x 2 + . . . + an x n ? Answer (a) 5 additions and 10 multiplications. (b) n additions and 2n multiplications.
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