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82 The Nuts and Bolts of Proof, Third Edition
Let us start by examining statements of the form "If A and B, then C."
Proving that such a statement is true does not require any special
technique, and some of these statements have already been included in
previous sections. The main characteristic of this kind of statement is that
the composite statement "A and B" contains several pieces of information,
and we need to make sure that we use all of them during construction of the
proof. If we do not, we are proving a statement different from the original.
Always remember to consider possible implicit hypotheses.
EXAMPLE 1. If fois a multiple of 2 and of 5, then fo is a multiple of 10.
Proof:
Hypothesis:
A. The number fc is a multiple of 2.
B. The number b is a multiple of 5.
{Implicit hypothesis: All the properties and operations of integer numbers
can be used.)
Conclusion:
C. The number fo is a multiple of 10.
By hypothesis A, the number fo is a multiple of 2. So, b = 2n for some
integer n. The other hypothesis, B, states that ft is a multiple of 5. Therefore,
b = 5k for some integer k. Thus,
2n = 5k.
Because 2n is divisible by 5, and 2 is not divisible by 5, we conclude that n is
divisible by 5. Thus, n = 5t for some integer number t. This implies that:
b = 2n = 2(50 = lOt
for some integer number t. Therefore, the number ft is a multiple of 10.
•
The proof of a statement of the form "If A and B, then C" can be
constructed using its contrapositive, which is "If'not C,' then either 'not A' or
'not B.'" (You might want to review the truth tables for constructing the
negation of a composite statement introduced in the How To Construct
the Negation of a Statement section.) This is a statement with multiple
conclusions which is part of the next topic presented.