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76 The Nuts and Bolts of Proof, Third Edition
Are all odd numbers larger than 2 prime numbers? The answer is
negative, because the number 9 is odd, but it is not prime. So, 9 G ^ but
9^B. Therefore, ylCR
Thus, the two sets are not equal. •
EXAMPLE 7. Let C = {all continuous functions on the interval [—1,1]
and D = {all differentiable functions on the interval [-1,1]. These two sets
are not equal.
Proof: All differentiable functions are continuous (a calculus book might
be helpful for checking this claim), but not all continuous functions are
differentiable.
Consider the function f{x) = \x\. This is continuous, but it is not
differentiable at x — 0. Thus, f e C but f^D. Therefore, CC D, and the
two sets are not equal. •
It is possible to define yet another operation between sets. Let A and B be
two subsets of the same universal set U. The difference set is the set:
A-B={aeA\a^B}.
EXAMPLE 8. Let A and B be two subsets of the same universal set U. The
following equahty holds true:
A-B
= AnB\
Proof: Let x e A — B.By definition this will happen if and only if x £ A
and x^B. This is equivalent to stating that x e A and x e B'. This will be
true if and only ii x e Ar^B'.
EXERCISES
Prove the following statements.
L For any three sets A, B, and C the following equahty holds:
(AU B)U {AU C) = AU {BU C).
2. The sets A = {all integer multiples of 2 and 3} and B = {all integer
multiples of 6} are equal.