1.3 ALGEBRAIC PROPERTIES OF SETS 35THEOREMS INVOLVING A CONDITIONALAs we progress in abstract mathematics, we find that the most interesting
theorems (and many interesting theorems of set theory in particular) do
not have so simple a form as those we have conjectured thus far. Instead of
the form "for any pair of sets X and Y, some relationship involving X and
Y is true," the more complex form "for every pair of sets X and Y, if
relationship (P) involving X and Y is valid, then so is relationship (Q) in-
volving the same two sets" occurs more often as we advance further into
the subject. The logical structure of statements will be the topic of Chapters
2 and 3, and so we will not be too technical at this point. Rather, we will
begin to explore this area by using examples and, especially, by recalling
Question (2) from the discussion following Example 3, earlier in this article.
EXAMPLE 4 Explain why the sets A, B, and D of Example 3 satisfy the
relationship (A n B) u D = A n (B u D).
Discussion In the paragraphs following Example 3 we found it to be
false that (X n Y) u Z = X n (Y u 2) for any three sets X, Y, and Z.
Thus we are led to inquire under what circumstances the preceding
equation will be valid, given that it is not always valid. We look for clues
to this from sets A, B, and D of Example 3. The equation (A n B) u D =
A n (B u D) was valid for these three particular sets. What was so spe-
cial about them? If we try to spot relationships (e.g., equality, subset,
etc.) between pairs of these sets, we note that D c A is the only such
relationship. Perhaps the previous equation turned out to be valid be-
cause D was a subset of A. Putting it differently, we are now wondering
whether we could find a subset X of A for which (A n B) u X does not
equal A n (B u X). We can test this by trying subsets X of {2,3,5,8)
other than (8) (which has already been tested). 4.
For instance, if X = (3,5), then (A n B) u X = {2,5) u (3,5) =
(2,3,5)=(2,3,5,8)n(1,2,3,5,6,7,1O)=An(BuX). Or,ifX=
{2,3,8), then (A n B) u X={2,5) u {2,3,8)={2,3,5,8)={2,3,5,8)
n {1,2,3,5,6,7,8, 10) = A n (B u X). In these two cases, in which
X is a subset of A, the equation (A n B) u X = A n (B u X), which is
not generally true for any three sets, holds true. We could continue
further, trying all possible subsets X of A and comparing (A n B) u X
with A n (B u X). (Do you know how many such subsets there are?
Counting problems such as this will be the subject of Article 1.5.) If we
were to exhaust all these cases (try a few more for your own benefit, say,
X = (3) or X = (2,8)), we would find that the equality of (A n B) u X
with A n (B u X) always holds.
Suppose then we want to state this officially as a conjecture. What is
an elegant way of saying that, for any sets A and B, there is no subset
X of A for which (A n B) u X does not equal A n (B u X)? In par-
ticular, we would like a formulation that avoids the double negative of