For each given statement, write (a) the converse, (b) the...

For each given statement, write (a) the converse, (b) the inverse, and (c) the contrapositive in if . . . then form. In some of the exercises, it may be helpful to restate the statement in if . . . then form.
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Key Concepts

Contrapositive

The contrapositive of a conditional statement is obtained by both reversing and negating the hypothesis and conclusion, resulting in the form 'if not q then not p'. Importantly, a conditional statement and its contrapositive are logically equivalent—that is, they always have the same truth value.

Inverse

The inverse of a conditional statement is formed by negating both the hypothesis and the conclusion of the original statement. This gives the statement 'if not p then not q'. The inverse is also not logically equivalent to the original conditional statement, though it often shares similar properties with the converse.

Converse

The converse of a conditional statement is created by reversing the hypothesis and the conclusion. In other words, if the original statement is 'if p then q', the converse is 'if q then p'. While the original statement and its converse are related in structure, they are not logically equivalent.

Conditional Statement

A conditional statement is a logical statement that follows the form 'if p then q', where p is the hypothesis and q is the conclusion. This type of statement establishes a relationship between two propositions, specifying that if the first is true, then the second must also be true.

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