Download A – B

Math in Our World
Section 2.2
Subsets and Set Operations
Learning Objectives
Define the complement of a set.
Find all subsets of a set.
Use subset notation.
Find the number of subsets for a set.
Find intersections, unions, and differences of
sets.
Find the Cartesian product of two sets.
Universal Set
Universal Set
A universal set, symbolized by U, is the set
of all potential elements under consideration
for a specific situation.
Once we define a universal set in a given setting, we
are restricted to considering only elements from that
set. If U = {1, 2, 3, 4, 5, 6, 7, 8}, then the only elements
we can use to define other sets in this setting are the
integers from 1 to 8.
Complement of a set, “A”
Complement
The complement of a set A, symbolized A,
is the set of elements contained in the
universal set that are not in A. Using setbuilder notation, the complement of A is
A = {x | x  U and x  A}.
U
A
A
This Venn Diagram shows the visual
representation of the sets U and A.
The complement of a set A is all the
things inside the rectangle, U, that
are not inside the circle representing
set A.
EXAMPLE 1
Finding the
Complement of a Set
Let U = {v, w, x, y, z} and A = {w, y, z}. Find A and
draw a Venn diagram that illustrates these sets.
SOLUTION
Using the list of elements
in U, we just have to cross
out the ones that are also
in A. The elements left
over are in A.
U = {v, w, x, y, z}
A = {v, x}
U
A
w z
y
v
x
Subsets
Subsets
If every element of a set A is also an
element of a set B, then A is called a subset
of B. The symbol  is used to designate a
subset; in this case, we write A  B.
•Every set is a subset of itself. Every element of a set
A is of course an element of set A, so A  A.
•The empty set is a subset of every set. The empty set
has no elements, so for any set A, you can’t find an
element of that is not also in A.
EXAMPLE 2
Finding All Subsets of a Set
Find all subsets of A = {American Idol, Survivor}.
SOLUTION
Number of elements in Subset Subsets with that Number of Elements
2
1
0
EXAMPLE 2
Finding All Subsets of a Set
Find all subsets of A = {American Idol, Survivor}.
SOLUTION
Number of elements in Subset Subsets with that Number of Elements
2
1
{American Idol, Survivor}
{American Idol}, {Survivor}
0

So the subsets are: {American Idol, Survivor},
{American Idol}, {Survivor}, 
Proper Subset
Proper Subsets
If a set A is a subset of a set B and is not
equal to B, then we call A a proper subset
of B, and write A  B.
The Venn diagram for a proper subset is shown below.
In this case, U = {1, 2, 3, 4, 5}, A = {1, 3, 5}, and
B = {1, 3}.
A
U
B
1 3
4
5
2
EXAMPLE 3
Finding Proper Subsets
of a Set
Find all proper subsets of {x, y, z}.
SOLUTION
Number of elements in Subset Subsets with that Number of Elements
3
2
1
{x, y, z}
{x, y}, {x, z}, {y, z}
{x}, {y}, {z}
0

We’ll eliminate
this one since
it’s equal to the
original.
So the proper subsets are: {x, y}, {x, z}, {y, z}, {x}, {y}, {z}, 
EXAMPLE 4
Understanding Subset Notation
State whether each statement is true or false.
(a){1, 3, 5}  {1, 3, 5, 7}
(b) {a, b}  {a, b}
(c) {x | x  N and x > 10}  N
(d) {2, 10}  {2, 4, 6, 8, 10} . - “not a subset of”
(e) {r, s, t}  {t, s, r}  - “not a subset of”
(f ) {Lake Erie, Lake Huron}  The set of Great
Lakes
EXAMPLE 4
Understanding Subset Notation
SOLUTION
(a) All of 1, 3, and 5 are in the second set, so {1, 3, 5} is a subset of
{1, 3, 5, 7}. The statement is true.
(b) Even though {a, b} is a subset of {a, b}, it is not a proper subset,
so the statement is false.
(c) Every element in the first set is a natural number, but not all
natural numbers are in the set, so that set is a proper subset of
the natural numbers. The statement is true.
(d) Both 2 and 10 are elements of the second set, so {2, 10} is a
subset, and the statement is false.
(e) The two sets are identical, so {r, s, t} is not a proper subset of {t,
s, r}. The statement is true.
(f ) Lake Erie and Lake Huron are both Great Lakes, so the
statement is true.
EXAMPLE 5
Understanding Subset Notation
State whether each statement is true or false.
(a)  {5, 10, 15}
(b) {u, v, w, x}  {x, w, u}
(c) {0}  
(d)   
EXAMPLE 5
Understanding Subset Notation
SOLUTION
(a) True: the empty set is a proper subset of every set.
(b) False: v is an element of {u, v, w, x} but not {x, w, u}.
(c) The set on the left has one element, 0. The empty
set has no elements, so the statement is false.
(d) The empty set is a subset of itself (as well as every
other set), but not a proper subset of itself since it is
equal to itself. The statement is false.
Number of Subsets for a Finite Set
If a finite set has n elements, then the set
has 2n subsets and 2n – 1 proper subsets.
Number of elements : n
Number of subsets : 2n
0
1
1
2
2
4
3
8
Number of proper subsets : 2n – 1
0
1
3
7
EXAMPLE 6
Finding the Number of
Subsets of a Set
Find the number of subsets and proper subsets of
the set {1, 3, 5, 7, 9, 11}.
SOLUTION
The set has n = 6 elements, so there are 2n, or 26 = 64,
subsets.
Of these, 2n – 1 , or 64 – 1 = 63, are proper.
Intersection of Sets
The intersection of two sets A and B,
symbolized by A  B, is the set of all
elements that are in both sets.
In set-builder notation,
A  B = {x | x  A and x  B}.
U
A
B
Note that the word “and”
means intersection.
The shaded area represents the intersection of sets A and B.
EXAMPLE 7
Finding Intersections
If A = {5, 10, 15, 20, 25}, B = {0, 10, 20, 30, 40},
and C = {30, 50, 70, 90}, find
(a) A  B (b) B  C (c) A  C
SOLUTION
(a) The elements 10 and 20 are in both sets A and B, so
A  B = {10, 20}.
(b) The only member of both sets B and C is 30, so
B  C = {30}.
(c) There are no elements common to sets A and C, so
A  C = .
Disjoint Sets
When the intersection of two sets is the
empty set, the sets are said to be disjoint.
U
A
B
For example, the set of students who stop attending
class midway through a term and the set of
students earning A’s are disjoint, because you can’t
be a member of both sets.
Union of Sets
The union of two sets A and B, symbolized
by A  B, is the set of all elements that are
in either set A or set B (or both).
In set-builder notation,
A  B = {x | x  A or x  B}.
U
A
B
Note that the word “or”
means union.
The shaded area represents the union of sets A and B.
EXAMPLE 8
Finding Unions
If A = {0, 1, 2, 3, 4, 5}, B = {2, 4, 6, 8, 10}, and
C = {1, 3, 5, 7}, find each.
(a) A  B (b) A  C (c) B  C
SOLUTION
To find a union, just make a list of all the elements in
either set without writing repeats.
(a) A  B = {0, 1, 2, 3, 4, 5, 6, 8, 10}
(b) A  C = {0, 1, 2, 3, 4, 5, 7}
(c) B  C = {1, 2, 3, 4, 5, 6, 7, 8, 10}
EXAMPLE 9
Performing Set Operations
Let A = {l, m, n, o, p}, B = {o, p, q, r}, and
C = {r, s, t, u}. Find each.
(a) (A  B)  C
(b) A  (B  C)
(c) (A  B)  C
EXAMPLE 9
Performing Set Operations
SOLUTION
A = {l, m, n, o, p}, B = {o, p, q, r}, and C = {r, s, t, u}
(a)(A  B)  C
First find A  B :
A  B = {l, m, n, o, p, q, r}.
Then intersect this set with set C.
The only common element is r, so (A  B)  C = {r}.
EXAMPLE 9
Performing Set Operations
SOLUTION
A = {l, m, n, o, p}, B = {o, p, q, r}, and C = {r, s, t, u}
(b) A  (B  C)
First find B  C :
B  C = {o, p, q, r, s, t, u}.
Then intersect this set with set A.
So A  (B  C) = {o, p}.
EXAMPLE 9
Performing Set Operations
SOLUTION
A = {l, m, n, o, p}, B = {o, p, q, r}, and C = {r, s, t, u}
(c) (A  B)  C
First find A  B :
A  B = {o, p}.
Then find the union of this set with set C.
So (A  B)  C = {o, p, r, s, t, u}.
EXAMPLE 10
Performing Set Operations
If U = {10, 20, 30, 40, 50, 60, 70, 80},
A = {10, 30, 50, 70}, B = {40, 50, 60, 70}, and
C = {20, 40, 60}, find each.
(a) A  C
(b)(A  B)  C
(c) B  (A  C)
EXAMPLE 10
Performing Set Operations
SOLUTION
U = {10, 20, 30, 40, 50, 60, 70, 80},
A = {10, 30, 50, 70} B = {40, 50, 60, 70} C = {20, 40, 60}
(a) A  C
First find A by eliminating the elements in set A from U.
So A = {20, 40, 60, 80}.
Then find C by eliminating the elements in set C from U.
So C = {10, 30, 50, 70, 80}.
Now note that 80 is the only element common to both, so
A  C = {80}.
EXAMPLE 10
Performing Set Operations
SOLUTION
U = {10, 20, 30, 40, 50, 60, 70, 80},
A = {10, 30, 50, 70} B = {40, 50, 60, 70} C = {20, 40, 60}
(b) (A  B)  C
First find A  B since it’s in parentheses.
So A  B = {50, 70}.
Then find (A  B) by eliminating the elements in set
A  B from U.
So (A  B) = {10, 20, 30, 40, 60, 80}.
Finally, we find the intersection of this set and C, so
(A  B)  C = {20, 40, 60}.
EXAMPLE 10
Performing Set Operations
SOLUTION
U = {10, 20, 30, 40, 50, 60, 70, 80},
A = {10, 30, 50, 70} B = {40, 50, 60, 70} C = {20, 40, 60}
(a)B  (A  C)
First find C by eliminating the elements in set C from U.
C = {10, 30, 50, 70, 80}
Then find A  C.
A  C = {10, 30, 50, 70}
Then find B by eliminating the elements in set B from U.
B = {10, 20, 30, 80}
So B  (A  C) = {10, 20, 30, 50, 70, 80}
Set Subtraction
The difference of set A and set B is the
set of elements in set A that are not in
set B. In set-builder notation,
A – B = {x | x  A and x  B}.
EXAMPLE 11
Finding the Difference of
Two Sets
Let U = {2, 4, 6, 8, 10, 12}, A = {4, 6, 8, 10},
B = {2, 6, 12}, C = {8, 10}
Find each.
(a) A – B
(b) A – C
(c) B – C
EXAMPLE 11
Finding the Difference of
Two Sets
SOLUTION
U = {2, 4, 6, 8, 10, 12}, A = {4, 6, 8, 10},
B = {2, 6, 12}, C = {8, 10}
(a) Start with the elements in set A and take out the
elements in set B that are also in set A. In this case, only
6 is removed, and A – B = {4, 8, 10}.
(b) Start with the elements in set A and remove the
elements in set C that are also in set A. In this case, 8
and 10 are removed, and A – C = {4, 6}.
(c) Start with the elements in set B and take out the
elements in set C that are also in set B. In this case,
none of the elements in B are also in C.
So B – C = {2, 6, 12}.
Cartesian Product
The Cartesian product (denoted A  B)
of two sets A and B is formed by writing all
possible ordered pairs in which the first
component is an element of A and the
second component is an element of B.
Using set-builder notation,
A  B = {(x, y) | x  A and y  B}.
Ordered Pair
An ordered pair is a pair of numbers or
objects that are associated by writing
them together in a set of parentheses, like
(3, 5). In this ordered pair, 3 is called the
first component and 5 is called the second
component.
EXAMPLE 12
Finding Cartesian Products
If A = {1, 3, 5} and B = {2, 4}, find A  B and B  A.
SOLUTION
To form A  B, first form ordered pairs with first
component 1: (1, 2) and (1, 4). Then form pairs with
first component 3: (3, 2) and (3, 4). Finally, use 5 as
the first component: (5, 2) and (5, 4).
A  B = {(1, 2), (1, 4), (3, 2), (3, 4), (5, 2), (5, 4)}.
For B  A, form all possible ordered pairs with first
components from B and second components from A:
B  A = {(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5)}.