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.
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