1.2 OPERATIONS ON SETS 19What do the last two parts of Example 4 suggest about the complement
of the complement of a set? Also, calculate C u D and C n D.... Are any
general facts suggested? Next, calculate (A n B)'. The use of parentheses
indicates that you are to perform the operation of intersection first, then
take the complement of the resulting set. Finally, calculate A n B', where
the lack of parentheses dictates that you first calculate the complement of
B, and then intersect that set with A.
EXAMPLE 5 Let U be the set of all employees of a certain company. Let
A = {XE Ulx is a male), B = {x E UIX is 30 years old or less}, C =
{x E U (X is paid $20,000 per year or less}. Describe the sets A n C, A',
A u B', and C n B'.
Solution A n C = {x E U I x is a male and x is paid $20,000 per year or less).
We might paraphrase this by saying that A n C consists of males who
make $20,000 per year or less. A' = {xlx is not a male) = {xlx is a
female). A u B' = {x leither x is a male or x is not 30 years old or less),
that is, the set of all male employees together with all employees over 30.
Finally, C n B' is the set of all employees over 30 years old who are paid
$20,000 per year or less. 0You should describe the sets A n B', (A n B)', (A u B)', A u A', and
C n C' in Example 5. See Exercise 12 for a mathematical example similar
in nature to Example 5.
SET THEORETIC DIFFERENCE
In introducing the operation of complement, we noted that the complement
of a set A is a relative concept, depending on the universal set as well as on
A itself. However, for a fixed universal set U, the complement A' of A de-
pends on A only. Our next operation on sets provides a true notion of
"relative complement." Set theoretic diflerence, denoted B - A, is a binary
operation that yields the complement of A relative to a set B.
DEFINITION 4
Let A and B bgsets. We define the difference B - A (read "6 minus A") by the
rule €3 - A = (x)x E 5 and x $ A). $The difference B - A (also called the complement of A in B, or the com-
plement of A relative to B) consists of all objects that are elements of B &
are not elements of A. If B = U, then B - A = U - A = A', the ordinary
complement of A. Thus complement is a special case of diference. Note also
that we need not know U in order to compute B - A.