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84 The Nuts and Bolts of Proof, Third Edition
EXAMPLE 2. Let x, y, and z be counting numbers. If x is a multiple of z
or y is a multiple of z, then their product xy is a multiple of z.
Proof:
Case 1. Let x be a multiple of z. Then x = kz with k integer (positive
because x > 0, z > 0). Therefore,
xy = (kz)y = (ky)z.
The number ky is a positive integer because k and y are positive integer.
So xy is a multiple of z.
Cas^ 2. Let 3; be a multiple of z. Then ); = nz with n integer (positive
because j; > 0, z > 0). Therefore,
xy = x(nz) = {xn)z.
The number xn is a positive integer because x and n are positive integer.
So xy is a multiple of z. •
The proof of a statement of the form "If A or B, then C" can be
constructed using its contrapositive, which is "If 'not C,' then 'not A' and
'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 again a statement with multiple
conclusions which is part of the next topic.
The contrapositive of the original statement in Example 2 is the
statement:
"Let X, y, and z be counting numbers. If the product xy is not a multiple
of z, then x is not a multiple of z and y is not a multiple of z"
MULTIPLE CONCLUSIONS
The most common kinds of multiple conclusion statements are:
L If A, then B and C;
2. If A, then B or C.
We will consider these statements in some detail.
L If A, then B and C.
The proof of this kind of statement has two parts:
i. If A, then B;
ii. If A, then C.