Math 141
Chapter 6 – Sets
Texas A&M – Summer 2014
Chapter 6 : Sets and Counting
Introduction to Sets
Sets
•
•
•
•
Set is a collection of items, referred to as its elements or members.
A set is represented by a capital letter.
For example, A = {1, 2, 3, 4, 5} is a set A containing elements 1, 2, 3, 4, and 5.
A set can be written in two notations:
• Roster Notation: explicitly writing all the elements. e.g. A = {1, 2, 3, 4, 5}
• Set-builder Notation: define the set in terms of its properties.
e.g. A = {x| x is a set of first five natural numbers}
READ AS : “the set of all x such that x is one of the first five natural numbers.”
“An element of” or “belongs to” (∈)
•
•
Given the set A = {1, 2, 3, 4, 5}
We can write 2 ∈ A and it is read as “2 is an element of set A.”, i.e. 2 is contained in set A.
Example: Write the set containing the letters in the word BADGE.
Example: Write the set containing the letters in the word MATHEMATICS.
1
Math 141
Chapter 6 – Sets
Texas A&M – Summer 2014
Subset
•
If every element of a set B is also in set A, then B is a subset of A, and is written as
BÍA
•
If every element in set B is also in set A, and B ¹ A, then B is a PROPER SUBSET of A and is
written as B Ì A
•
•
Every set is a subset of itself.
Empty set is a subset of every set.
Empty and Universal Set
•
Empty Set: A set with NO ELEMENTS. Represented as { } or Æ.
•
Universal Set: Set of ALL elements being considered. It's denoted by U.
Example : Let A = {1, 2, 3, 4}
B = {2, 4, 3, 1}
State whether each of the following is True or False.
a) B Í A
b) C Í A
c) C Ì A
d) B Ì A
e) A Í A
f) 2∈B
g) {4, 1} Ì C
h) 6 Ï B
i ) {2} ∈ B
2
C = {1, 3}
Math 141
Chapter 6 – Sets
Texas A&M – Summer 2014
Example Given the set M = {a, b, c}.
a) List all the subsets of the set M.
b) List all the proper subsets of the set M.
c) How many subsets does M have?
d) How many PROPER subsets does M have?
Example State whether the following are True or False, given T = { 1, 3, 5, 7, 9}
a) T = {x| x is a set of positive odd numbers less than 10}
b) If S = {x| x is a set of prime numbers greater than 0 and less than 10}, then S =T.
c) S È T has 15 proper subsets.
c) {11} ∈ T
d) {7, 9, 13} Í T
e) Æ Í T
3
Math 141
Chapter 6 – Sets
Texas A&M – Summer 2014
Set Operations
Consider the sets: U = { 1, 2, 3, 4, 5, 6, 7, 8 }
A = { 1, 2, 3 } ;
C = { 5, 7 }
B = { 2, 4, 6 } ;
D = { 3, 2, 1 } ;
E = { 2, 3, 4, 5 }
1. Set Union (Recall OR from the Chapter Logic):
• Union of two sets is a set of all elements that belong to A, or B, or both.
• In other words, Union of two sets is a set of all elements that belong to ATLEAST ONE of A or
B.
• It's represented as A Ç B.
2. Set Intersection (Recall AND from the Chapter Logic):
• Intersection of two sets is a set of all elements that belong to BOTH, A and B.
• It's represented as
• Two sets A and B are disjoint if they have no elements in common, i.e. if A È B = Æ.
3. Complement of a Set (Recall NOT from the Chapter Logic):
• Given a set A. The complement of the set A is the set of all the elements that are in set U, but
not in A.
• It's represented as AC
4
Math 141
Chapter 6 – Sets
Example Given the sets U, A, B, and C, find the indicated sets:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4, 5}
B = {2, 4, 6, 8}
C = {1, 3, 5, 9}
a) (A Ç B )C
b) ( B Ç C ) È A C
c) A È (B C Ç C )
d) AC È B È C C
e) B Ç B C
f) C È C C
5
Texas A&M – Summer 2014
Math 141
Chapter 6 – Sets
Venn Diagrams:
• Universal set is represented by a rectangle.
• All other sets are usually represented by circles.
a) A Ç B C
b) ( B Ç C ) È A C
c) A È (B C Ç C )
d) ACÈ B È C C
e) B Ç B C
f) C È C C
6
Texas A&M – Summer 2014
Math 141
Chapter 6 – Sets
Laws of Set Operation (Can be verified using Venn Diagrams)
Commutative Laws : A Ç B = B Ç A
AÈB = BÈA
Associative Laws : A Ç (B Ç C ) = ( A Ç B ) Ç C
A È (B È C ) = ( A È B ) È C
Distributive Laws : A Ç (B È C ) = (A Ç B ) È ( A Ç C )
A È (B Ç C ) = (A È B ) Ç ( A È C )
De Morgan's Laws : ( A Ç B )C = ACÈ BC
( A È B )C = A C Ç B C
Proof of De-Morgan's Laws
a) ( A Ç B )C = ACÈ BC
b) ( A È B )C= AC Ç B C
7
Texas A&M – Summer 2014
Math 141
Chapter 6 – Sets
Texas A&M – Summer 2014
Union Rule or Addition Rule of Sets
If A and B are two sets such that A Ì U and B Ì U, and A and B are not disjoint then
n(A Ç B) =
Proof :
If A and B are disjoint sets such that A Ì U and B Ì U, then
n(A Ç B) =
Problems
1. If n(B) = 100, n(A Ç B) = 175, n(A È B) = 40, what is n(A), n(A È B)C , n(BC) , n( ACÇ BC).
2. If n(U) = 200, n(A Ç B) = 150, what is n( AC È BC ) ?
8
Math 141
Chapter 6 – Sets
Texas A&M – Summer 2014
3. If n(B) = 200, n(A È B È C ) = 40, n(A È B È C' ) = 20, n(AC È B È C ) = 50, what is
n(AC È B È C C ).
4. In a survey of 500 people, a pet food manufacturer found that 200 owned a dog but not a cat, 150
owned a cat but not a dog, and 100 owned neither a dog or cat.
a) How many owned both a cat and a dog?
b) How many owned a dog?
9
Math 141
Chapter 6 – Sets
Texas A&M – Summer 2014
5. A survey of 600 people over the age of 50 found that 200 owned some stocks or real estate but no
bonds, 220 owned some real estate or bonds but no stock, 60 owned real estate but no stocks or bonds,
and 130 owned both stocks and bonds. How many owned none of the three?
10
Math 141
Chapter 6 – Sets
Texas A&M – Summer 2014
6. a survey of 500 people found that 190 played golf, 200 skied, 95 played tennis, 100 played golf but
did not ski or play tennis, 120 skied but did not play golf or tennis, 30 played golf and skied but did not
play tennis, and 40 did all three.
a) How many played golf and tennis but did not ski?
b) How many played tennis but did not play golf or ski?
c) How many participated in at least one of the three sports?
d) How many did not play any or the sport?
e) How many participated in exactly one sport?
f) How many play exactly two of these sport?
11