180 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5
(b) THEOREM For any sets A and B, A' - B' = B - A.
"Proof" Let A and B be arbitrary Sets. Then
A' - B' = A' n B" [Fact 5 (29), Article 1.41
= A' n B [Fact 3 (21), Article 1.41
= B n A' [Fact 2 (16), Article 1.41
= B - A [Fact 5 (29), Article 1.41(c) FACT If f(x) = x - (llx), then f is not an even function. (cf; Exercise 9).
Start of "Proof" Let x be an arbitrary nonzero real number. To prove f is not even,
we must prove that f (-x) # f (x)....
(d) THEOREM For any sets A and B, (A - 6)' = A' u 6.
Start of "Proof" Let A and B be arbitrary sets. If (A - B)' = A' u B, then
(AnB')'=A'u B....Conclusions Involving V and +, but Not
In this article we consider the problem of proving statements whose con-
clusion has the logical form (Vx)(p(x) -, q(x)). Such proofs are common, for
many definitions in mathematics have this logical form. Here are a few
examples:EXAMPLE 1 A subset C of R x R is said to be symmetric with respect to
the x axis (respectively, y axis and origin) if and only if, for all real
numbers x and y, (x, y) E C implies (x, - y) E C [respectively, (- x, y) E C
and (-x, -y) E C].Observe that the logical form of the definition of x-axis symmetry in
Example 1 is (Vx)(Vy) (Ax, y) -* q(x, y)), where Ax, y) is the predicate
(x, y) E C and q(x, y) represents (x, - y) E C.EXAMPLE 2 A subset I of the set of all real numbers R is said to be an
interval if and only if, for all a, b, c E R, if a E I, c E I, and a c b c c,
then b E I (recall befinition 2, Article 1.1).EXAMPLE 3 A real-valued function y = f(x) is said to be increasing on
an interval I if and only if, for all x, and x2 E I, if x, < x2, then
f (XI) < f (~2)-EXAMPLE 4 A set A is said to be a subset of a set B (A and B both con-
tained in a common universal set U) if and only if, for every x E U, x E A
implies x E B [recall Definition l(b), Article 4.11.