epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Special Kinds of Theorems 87
of the possible conclusions is true; that is, if one of the two conclusions is
false, then the other must be true. Thus, the best way to prove this kind of
statement is to use the following one, which is logically equivalent to it:
If A and (not B), then C.
It might be useful to consider the truth tables for the two statements
"If A, then B or C" and "If A and (not B), then C."
A
B
C
BorC
If A, then B or C
T
T
T
T
T
T
T
F
T
T
T
F
T
T
T
T
F
F
F
F
F
T
T
T
T
F
T
F
T
T
F
F
T
T
T
F
F
F
F
T
A
B
C
notB
A and (not B)
If A and (not B), then C
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
F
F
T
T
F
F
T
T
F
F
T
T
F
F
F
F
T
T
T
F
T
T
T
T
Similarly one can prove that the statements "If A, then B or C" and "If A and
(not C), then B" are logically equivalent.
EXAMPLE 5. Let n be a composite number larger than 1. Then n has at
least one nontrivial factor smaller than or equal to v^.
Discussion: We have:
A. The number n is a composite number larger than 1.
Thus, n = pq with 1 < p <n and I < q <n.