1.3 ALGEBRAIC PROPERTIES OF SETS 31COMMUTATIVITY AND ASSOCIATIVITYEXAMPLE2 LetU=(1,2,3 ,..., 9,lO),A={2,3,5,8),B=(1,2,5,6,
7, 101, and C = (2,3,4,9, 10). Compute(A n B) n Cand A n (B n C).
Solution The significance of parentheses is the signal they give to perform
the operation inside them first, then operate further with the result of
that first computation. Thus, to compute (A n B) n C, we first com-
pute A n B = {2, 51, and then calculate the intersection of this result
with C = {2,3,4,9, 10) to get (2). On the other hand, A n (B n C) is
obtained by computing B n C = (2, 10) and then intersecting this set
with A to get (2). In this particular example A n (B n C) = (2) =
(A n B) n C.
To test whether the equality of A n (B n C) with (A n B) n C for the
three particular sets given in Example 2 was accidental (i.e., dependent
on some special property of the given sets A, B, and C), or whether this
equation might represent a candidate for a theorem (i.e., be true for any
three sets), we construct two Venn diagrams as shown in Figure 1.7.
These two diagrams were drawn independently of each other and by
two different procedures. In the first diagram we shaded horizontally the
region corresponding to B n C and then shaded circle A vertically; in the
second diagram we shaded the A n B region and circle C. Yet the crucial
region in both pictures (i.e., the "crosshatched" region) is the same in both
diagrams, namely, the region common to all three circles. These pictures
support the case that the equality of Example 2 represents a general property.
You should be able to formulate further conjectures, based on the sets
A, B, and C of Example 2, by computing the sets A n B, B n A, B u C,
C u B, A u (B u C), and (A u B) u C and by constructing Venn diagrams
corresponding to these sets. After carrying out these exercises, you should
be ready to state Conjecture 2.
Figure 1.7 Venn diagrams suggesting Conjecture 2( f ).