Contrapositive and Converse

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CONTRAPOSITIVE AND CONVERSE

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DEFINITION: The converse of the sentence " If A, then B " is the sentence " If B, then A ".
Note that the converse switches the hypothesis and conclusion.

DEFINITION: The contrapositive of the sentence " If A, then B " is the sentence " If ()not B), then ()not A) ".
Note that the contrapositive negates the conclusion, and makes it the hypothesis.
It also negates the hypothesis, and makes it the conclusion.

Here are the truth tables for an implication, its contrapositive, and its converse:

A	B	not A	not B	an implication If A, then B	the contrapositive of the implication If ()not B), then ()not A)	the converse of the implication If B, then A
T	T	F	F	T	T	T
T	F	F	T	F	F	T
F	T	T	F	T	T	F
F	F	T	T	T	T	T

An analysis of these truth tables shows the following:

An implication is equivalent to its contrapositive.
Thus, the sentences " If A, then B " and " If ()not B), then ()not A) " are completely interchangeable:
if one is true, so is the other; if one is false, so is the other.
Mathematicians routinely find the contrapositive of an implication, to see if it is easier to work with than the original implication.
An implication is NOT equivalent to its converse.
Thus, the sentences " If A, then B " and " If B, then A " are not interchangeable.
The truth of each sentence must be investigated separately.

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