1.1. Sets and Functions 3

A B A B

A

B

A

B

Figure 1.1.1. Intersection and Union of Two Sets.

There is one special set that is a subset of every set. This is the empty set ∅.

It is the set with no elements. Since it has no elements, the statement that “each

of its elements is also an element of A” is true no matter what the set A is. Thus,

by the definition of subset,

∅ ⊂ A

for every set A.

If A and B are sets, then the intersection of A and B, denoted A ∩ B, is the

set of all objects that are elements of A and of B. That is,

A ∩ B = {x : x ∈ A and x ∈ B}.

Similarly, the union of A and B, denoted A ∪ B, is the set of objects which are

elements of A or elements of B (possibly elements of both). That is,

A ∪ B = {x : x ∈ A or x ∈ B}.

Example 1.1.2. If A is the closed interval [−1,3] and B is the open interval (1,5),

describe A ∩ B and A ∪ B.

Solution: A ∩ B = (1,3] and A ∪ B = [−1,5).

If A is a (possibly infinite) collection of sets, then the intersection and union of

the sets in A are defined to be

A = {x : x ∈ A for all A ∈ A}

and

A = {x : x ∈ A for some A ∈ A}.

Note how crucial the distinction between “for all” and “for some” is in these defi-

nitions.

The intersection A is also often denoted

A∈A

A or

s∈S

As

if the sets in A are indexed by some index set S. Similar notation is often used for

the union.