Bridge to Abstract Mathematics: Mathematical Proof and Structures

1.2 OPERATIONS ON SETS 21

in B, or in B but not in A. Stated differently, elements of A A B are objects
in one or the other of the sets A and B, but not in both.

EXAMPLE 7 Let A = {2,4, 6,8, 101, B = (6, 8, 10, 121, C = (1, 3, 5, 7,9,
111, and D = (4,6,8}. Calculate A A B, A A C, and A A D.

Solution A - B = {2,4) and B - A = (121, so that (A - B) u (B - A) =

AAB= (2,4,12). Similarly, AAC= {1,2 ,..., 11) and AAD=

(2,101. You should calculate B AA, A A(B A C) and (A A B) A C.

Are any possible general properties of the operation "symmetric differ-

ence" suggested by any of these examples?

EXAMPLE 8 Let W = (- oo, 3), X = (- 3,5], and Y = [4, oo). Compute

WAX and WA Y.

Solution WAX =(-a, -33 u [3,5] and WA Y =(-c0,3) u [4, a).

You should calculate X A Y.

ORDERED PAIRS AND THE CARTESIAN PRODUCT

The sixth and final operation on sets to be introduced in this article, carte-
sian product of sets, differs from the preceding five in a subtle but important
respect. If U were the universal set for sets A and B, it would again be the
universal set for A n B, A u B, A', A - B, and A A B. Putting it differently,
the elements of these sets are the same types of objects as those that consti-
tute A and B themselves. This is not the case for A x B, the cartesian prod-
uct of A and B. The elements of A x B are ordered pairs of elements from
A and B, and thus are not ordinarily members of the universal set for A and
B.
Ordered pairs resemble notationally two-element sets but differ in two
important respects. Represented by the symbol (a, b), the "ordered pair
a comma b" differs from the set (a, b} in that the order in which the elements
are listed makes a difference. Specifically, the ordered pairs (a, b) and (b, a)
are different, or unequal, unless a = b, based on Definition 6.

DEFINITION - 6

Given ordered pairs (a, 6) and (c, d), we say that these ordered pairs are

equal, denoted (a, b) = (c, d), if and only if a = c and b = d.

Compare this definition with the criterion for equality of the sets {a, b)

and {c, d) (Remark 2, Article 1 .I), and note, for example, that (2, 3) # (3,2),

whereas {2,3} = {3,2}. A second major distinction between ordered pairs

and two-element sets is that the same element may be used twice in an

ordered pair. That is, the expression (a, a) is a commonly used mathematical

symbol, but (a, a) is not (i.e., the latter is always expressed (a).)