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Survey of Math - MAT 140
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Venn Diagrams and Set Operations
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1.1
Operations on Sets
Compliment
Compliment of a Set - all the elements in the Universe that are NOT in the given set.
Notation A0 in English "A's Compliment"
Example 1 Given that
U D fx N jx < 11g and
Find A0 and write in Roster Form
A D fx 2 N jx < 10 and x is odd}
A D f1; 3; 5; 7; 9g
so all the elements that are not in A but are still in the Universe will be:
A0 D f2; 4; 6; 8; 10g
Example 2 Given
A D fx 2 f amiliesjx are families with at least one girlg nd A0 (the compliment of Set A)
We need to determine what types of elements are not in the above set. What sort of families Do Not Have "at least one
girl". So this must be the families that have only boys. Thus we can write:
A0 D fx 2 f amiliesjx are families with only boysg
1.2
Intersection
The Intersection of Two Sets will be the elements that the two sets have in common.
Symbolically this is written as A \ B which represents : "A intersect B" or "all the elements that A and B have in
Common"
Example 3 Given that:
A D f1; 2; 5; 7; 9; 11; 15; 17g and
B D f2; 4; 9; 17; 22g Find A \ B:
To nd this all we need to do is list down what A and B share in common. I nd it easiest to go through each element
of the rst set and ask if that element is in the other. If it is I write it down as part of my answer. Example: Is 1 in B?
no
Is 2 in B? yes , write it down and eventually arrive at the solution:
A \ B D f2; 9; 17g
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
1.3
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Union
Union-the combination of all the elements of two sets. Symbolical written as A [ B
Example 4 Given
A D f1; 4; 7; 10g and
B D f2; 4; 6; 8; 10g nd A [ B
English "A union B" or "list all elements of A and all the elements of B"
This becomes {1,4,7,10,...} from the Set A and then we include all the elements of B that are not already listed, which
in order will be {1,2,4,6,7,8,10}.
*Note that I did not repeat the numbers that were in common, they only need to be listed once.
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Venn Diagrams
Venn Diagrams are pictorial representation of Sets. The General set up of a Two Set environment is depicted below.
This represents all the Possibilities between two Sets A and B.
A
B
I
II
III
U
IV
If A D B, then Sector II is the only one with elements
If A and B are DISJOINT then sector II is empty, and I and III have elements we can say that .
If B
A, then sectors I and II are the only sectors with elements
Note: Sector IV is the place where elements that do not belong to either A or B are put. Here is where knowing
the Universe becomes important.
To construct a Venn Diagram it is usually wise to start in Sector II (the middle) and work your way out.
Example 5 Construct a Venn Diagram for the given sets.
U D fx 2 N jx
10g
A D fx 2 N jx is oddg
B D fx 2 N jx is a prime numberg
Start with what A and B have in common
A D f1; 3; 5; 7; 9g
B D f2; 3; 5; 7g
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 3
so: A \ B D f3; 5; 7g this will go in Sector II.
Next ll in Sector I what hasn't been used yet in A f1; 9g
and what hasn't been used for B in Sector III f2g
Numbers that are used in neither of the Sets will be put in Sector IV. f4; 6; 8; 10g and we get:
A
B
3
5
7
1, 9
2
4, 6, 8, 10
Now to use this picture to answer some question becomes easy. I will teach this using the Sector Method of Venn
Diagrams.
A
B
I
U
II
III
IV
Example 6 Using above data, nd:
1. A \ B
2. A0 \ B
3. A \ B 0
4. .A \ B/0
Answers: Using the sector notation we get that:
1. A \ B is sector II f3; 5; 7g
2. A0 D I I I; I V and B D I I; I I I so what they share in common is I I I which has f2g
3. A D I; I I and B 0 D I; I V so what they share in common is I which has f1; 9g
4. A \ B is sector II so .A \ B/0 will be I; I I I; I V f1; 9; 2; 4; 6; 8; 10g
in order f1; 2; 4; 6; 8; 9; 10g
Example 7 All together now, lets design a Venn Diagram Set and then Answer the following questions based on the
Venn Diagram
U D fx 2 N jx < 16g
A D f2; 3; 5; 7; 11; 13g
B D f1; 3; 6; 7; 9; 10; 13; 15g
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
Page: 4
1) Start with what elements Set A and Set B have in common: A \ B D f3; 7; 13g
and put this in Sector II
2) Then what is left for the rest of A f2; 5; 11g in Sector I
(only write each element only ONCE for the whole Venn Diagram!!!!!!)
3) and in the rest of B f1; 6; 9; 10; 15g into Sector III
4) What ever is left for the Universe, that is not already in A or B will go into
Sector IV. We have: f4; 8; 12; 14g
A
B
2, 5, 11
3
7
13
1, 6, 9
10, 15
4, 8, 12, 14
Now lets nd:
1. A [ B
2. A0 [ B 0
3. A \ B 0
4. .A [ B/0
Answers: We will again use the Sector Method from before:
A
B
I
U
II
III
IV
1. A[B D I; I I; I I I which have the elements {2,5,11,3,7,13,1,6,9,10,15} in order f1; 2; 3; 5; 6; 7; 9; 10; 11; 13; 15g
2. A0 D I I I; I V and B 0 D I; I V so A0 [B 0 D I; I I I; I V {2,5,11,1,6,9,10,15,4,8,12,14} in order f1; 2; 4; 5; 6; 8; 9; 10; 11; 12; 14; 1
3. A \ B 0 D I is the only sector that they share in common f2; 5; 11g
4. .A [ B/0 D .I; I I; I I I /0 D I V so...f4; 8; 12; 14g
Copyright 2007 by Tom Killoran
Survey of Math - MAT 140
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Number of Elements in a Union of Two Sets
If n .A/ D 5 and n .B/ D 4; then does that mean that n .A [ B/ D 9? Remember that n .A/ is the number of elements
in a set.
Let
A D f1; 2; 3; 4; 5g
B D f4; 5; 6; 7g
then A [ B D f1; 2; 3; 4; 5; 6; 7g so n .A [ B/ D 7 NOT 9
So what happened?
Well two of the elements where shared by both of the sets, thus if we just added the number of elements in A and the
number of elements in B we would have counted f4; 5g twice.
Formula: n .A [ B/ D n .A/ C n .B/
n .A \ B/ (takes away the number they have in common)
Copyright 2007 by Tom Killoran