Math in Our World
Section 2.2 D2
Subsets and Set Operations
Learning Objectives
Find intersections and unions
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.
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).
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}.
Classwork
p. 63-64: 45-51 all, 55-58 all, 60, 62, 65-70 all, 75-79
all, 81, 82