Download An “If–Then” Statement and Its Contrapositive: A Useful Rule in

An “If–Then” Statement and Its Contrapositive: A Useful Rule in Logic
Suppose you have a statement of the form
“If X , then Y”
(1)
(where X and Y are themselves statements—about mathematical things or about anything else). A mathematical example would be
“If 6 divides some number x, then 3 divides the number x.”
(2)
Two nonmathematical examples would be
“If that restaurant is open, then the lights inside it will be on.”
(3)
“If my car won’t start, then the battery is dead.”
(4)
and
The contrapositive of such a statement is the statement you get by doing BOTH of two things: you must
exchange the X and the Y AND you must negate both X and Y. The contrapositive of (1) is
“If not Y, then not X .”
(1C)
We will construct the contrapositives of (2), (3) and (4) together in class:
(2C)
(3C)
(4C)
The reason for thinking about contrapositives is that (1) (1C) always have the same truth value; that is,
if (1) is true (respectively false), then (1C) is also true (respectively false). Below I explain why this is so in
general, but let’s first check this together on examples (2), (3) and (4).
First Property. If
“If X , then Y”
(1)
is true, then
“If not Y, then not X ”
(1C)
is also true.
Proof. We are assuming that (1) is true. Say that “not Y” is true, so that Y is false. If X were true, then
by (1), Y would also be true, which it isn’t, so X must be false, so that “not X ” must be true.
Second Property. If
“If not Y, then not X ”
(1C)
is true, then
“If X , then Y”
(1)
is also true.
Proof. We are assuming that (1C) is true. If we apply the first property to (1C), we get
“If not not X then not not Y;”
(1CC)
but “not not X ” means the same thing as X and “not not Y” means the same thing as Y, so (1CC) means
the same thing that (1) does.
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This is useful for two reasons. First: whenever you prove a statement of the form “If X , then Y”—and,
broadly speaking, everything you prove is of this form—then you get a freebie, because you have also proved
“If not Y, then not X .” Second: if you are trying to prove a statement of the form “If X , then Y,” it
sometimes helps to try instead to prove the contrapositive statement “If not Y, then not X .”
Exercise 1. For each of the statements below, write down the contrapositive statement, eliminating any
occurrences of “not not.” Then state whether the pair of statements is true or false.
[a]: If n is odd, then 2 does not divide n.
[b]: If n is even, then 2 does not divide n.
[c]: If that car was manufactured before 1995, then it is not an electric car.
[d]: If n is prime, then n is odd.
[e]: If n is odd, then n is prime.
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