Page 154 - Discrete Mathematics and Its Applications

2.2 Set Operations 133

EXAMPLE 15 Let A ={0, 2, 4, 6, 8}, B ={0, 1, 2, 3, 4}, and C ={0, 3, 6, 9}. What are A ∪ B ∪ C and
A ∩ B ∩ C?

Solution: The set A ∪ B ∪ C contains those elements in at least one of A, B, and C. Hence,

A ∪ B ∪ C ={0, 1, 2, 3, 4, 6, 8, 9}.

The set A ∩ B ∩ C contains those elements in all three of A, B, and C. Thus,

A ∩ B ∩ C ={0}. ▲

We can also consider unions and intersections of an arbitrary number of sets. We introduce
these definitions.

DEFINITION 6 The union of a collection of sets is the set that contains those elements that are members of
at least one set in the collection.

We use the notation

n

A 1 ∪ A 2 ∪ ··· ∪ A n = A i
i=1

to denote the union of the sets A 1 ,A 2 ,...,A n .

DEFINITION 7 The intersection of a collection of sets is the set that contains those elements that are members
of all the sets in the collection.

We use the notation

n

A 1 ∩ A 2 ∩ ··· ∩ A n = A i
i=1

to denote the intersection of the sets A 1 ,A 2 ,...,A n . We illustrate generalized unions and
intersections with Example 16.

EXAMPLE 16 For i = 1, 2,..., let A i ={i, i + 1,i + 2,... }. Then,

n n

A i = {i, i + 1,i + 2,... }={1, 2, 3,... },
i=1 i=1

and

n n

A i = {i, i + 1,i + 2,... }={n, n + 1,n + 2,... }= A n .
▲
i=1 i=1