6.1
Sets
A set is a collection of objects with a clear definition. We must always be able to tell what belongs to the set. The objects in the set are called elements.
Two ways to clearly define a set are :
1) Roster notation—the elements are listed within brackets
For example A={1,2,3,4,5,6,7,8,9,10} which can also be written A={1,2,3,…,10}.
2) Set builder notation—the elements are described
Example: A above can be written as A={x| x is an integer between 1 and 10 inclusive}
Or as A={x| x is an integer and 1<x<10} a 
A means a is an element of set A. a 
A means a is not an element of set A.
{1} is a singleton set. It has one element, the number 1. {2}, {b}, {A} are also singleton sets, even if A is a set with many elements.
For A as above, 1  A is true. {1}  A is also true because the set {1} is not in the list of elements of A.
The empty set is written  , and is the set with no elements.
A  B means A is contained in B, A is a subset of B. Every element of A is an element of B.
B  A means the same thing.
A  B means A is properly contained in B. A is contained in B and B has at least one element not in A.
The number of subsets of an n-element set is 2 n .
Why is this so? The only subset of the empty set is the empty set. If A ={1}, the subsets of A are the empty set and {1}. If B= {1,2} the subsets of B are  , {1}, {2}, and {1,2}. How many subsets of a 3-element set are there? If C={1,2,3}, the subsets are  , {1}, {2}, {1,2} which are the subsets of B and {3}, {1,3}, {2,3}, and {1,2,3} which are each of the subsets of B with 3 added in. So the number of subsets doubled. Each time we add a new element we get all the old subsets and all these with the new element thrown in. So the number of subsets doubles each time we add a new element.
Example: If A={1,2,3,4,5,6,7,8,9,10} and B={1,2,3,…}={x| x is a positive integer} then
A  B and A  B are both true. A  A is always true.
We say A is a proper subset of B. A is not a proper subset of A.
The universal set, U, is the set of all elements under consideration.
Ex. A={x| x is a math 141 student at TAMU} B={x| x is an accounting major at TAMU}. A suitable universal set is U={x| x is a student at Tamu}.
A={x| x  U, x is a math 141 student} B={x| x  U, x is an accounting major}
A  B={x| x  A or x  B} is the union of A and B. The or allows the possibility that x is in both A and B. A union B is the set of x in at least one of A and B.

included.
A  B={x| x  A and x  B} is the intersection of A and B. The intersection is the set of
A

A

B , B
Ex. A={1,3,5,7} B={1,2,3,4} then A  B={1,2,3,4,5,7}. Notice 1 and 3 are elements in both A and B.
Ex. For A and B as above A

A

B , A

B

 B={1,3}
AandA

B

B are all true.
A and B are called disjoint if A  B= 
.
If C={4,6} and A is as above then A and C are disjoint.
A C ={x| x  U, x  A} is the complement of A. U is the universal set.
Ex. U={x| x is a positive integer} A={x| x is an even positive integer}
A C ={x| x is an odd positive integer}
Ex. U={x| x is a letter of the English alphabet} A={a,b,c,d,e,f}
A C ={ g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}
DeMorgan’s Laws
( A

B )
C 
A
C 
B
C If x is not in at least one of A and B then x is not in A and x is not in B.
( A

B )
C 
A
C 
B
C If x is not in both A and B then x is not in A or x is not in B.
Describing with words:
A

B

C “x is in at least one of A,B, and C”
A

B

C “x is in all three of A, B and C”
A

B
C 
C
C “x is in A only”
Example:
U is the set of people in College Station who shop at HEB, Kroger, Sam’s or nowhere.
H is the set of people in U who shop at HEB. K is the people in U who shop at Kroger and S is the people in U who shop at Sam’s.
Describe in terms of H, K, and S the sets of people who: shop at Sam’s only. shop in at least one of the three stores. shop at all three stores. shop at HEB or Sam’s but not Kroger.
6.2
The number of elements in a finite set. We restrict our discussion to finite sets. n(A) is the number of elements in A.
Ex. If A={a,b,c,d,e} n(A)=5. n(  )=0.
The inclusion-exclusion rule is n ( A

B )
 n ( A )
 n ( B )
 n ( A

B )

Ex: A={1,3,5,7} B={2,3,4,5,6} n(A)=4 and n(B)=5
A  B={1,2,3,4,5,6,7} A  B={3,5}
7=5+4-2
If we count A we count 1,3,5 and 7. If we count B we count 2,3,4,5, and 6.
Now 3 and 5 have been counted twice. A  B is counted twice so we have to subtract it once. n ( A

B

C )
 n ( A )
 n ( B )
 n ( C )
 n ( A

B )
 n ( A

C )
 n ( B

C )
 n ( A

B

C )
the inclusion- exclusion formula for three sets.
Why is the triple intersection added?
The triple intersection is included in A, B, and C. It is added in 3 times when we count each of these sets. The triple intersection is also included in each double intersection. It is subtracted 3 times when we subtract the numbers for each of these sets. Now we haven’t counted it at all so we have to add it back.
Ex. A company makes apple, banana and cherry fruit spreads. A survey of 250 people finds 140 use apple, 125 use banana, 125 use cherry, 85 use both apple and banana, 80 use both banana and cherry and 75 use both apple and cherry. 50 use none. How many use all three?
250-50=200 use at least one of the three. Using the inclusion-exclusion formula we have
200=140+125+125-85-80-75+n(
A

B

C )
so the number who use all three is 50.
Ex. This problem uses a different approach. 250 people were asked if they drink coffee tea or milk.
220 said they drink at least one of the three.
90 said they drink at least two of the three.
130 drink exactly one
50 drink only coffee
100 drink milk
120 drink tea
35 drink coffee and milk
45 drink only tea
Fill in each region of the Venn diagram. How many drink exactly two?