Section 4.1 Sets
233
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PraCtiCe 16 Let
A = {1, 2, 3, 5, 10}
B = {2, 4, 7, 8, 9}
C = {5, 8, 10}
be subsets of S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find
a. A c B
b. A − C
c. B′ d (A c C  )
We will define one final operation using elements of ℘(S).
Definition CaRteSiaN PROdUCt
Let A and B be subsets of S. The Cartesian product (cross product) of A and B,
denoted by A × B, is defined by
A × B = {(x, y) 0 x [ A and y [ B}
Thus, the Cartesian product of two sets A and B is the set of all ordered pairs
whose first component comes from A and whose second component comes from B.
The cross product is not a binary operation on ℘(S). Although it acts on an ordered pair of members of ℘(S) and gives a unique result, the resulting set is not,
in general, a subset of S. The elements are not members of S but ordered pairs of
members of S. So the resulting set is not a member of ℘(S). The closure property
for a binary operation fails to hold.
Because we will often be interested in the cross product of a set with itself, we
will abbreviate A × A as A
2
; in general, we use A
n
to mean the set of all ordered
n-tuples (x 1 , x 2 , … , x n ) of elements of A.
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PraCtiCe 17 Let A = {1, 2} and B = {3, 4}.
a. Find A × B.
b. Find B × A.
c. Find A
2
.
d. Find A
3
.
Set identities
There are many set equalities involving the operations of union, intersection,
difference, and complementation that are true for all subsets of a given set S.
Because they are independent of the particular subsets used, these equalities are called set identities. Some basic set identities follow. The names and
forms of these identities are very similar to the tautological equivalences of
Section 1.1 (check back and compare). We will see in Chapter 8 that this
similarity is not a coincidence.