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.
$p \rightarrow \sim q$

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Key Concepts

Conditional Statement

A conditional statement, also known as an 'if-then' statement, is a logical statement that connects two propositions. The first part (antecedent) leads to the second (consequent), and it asserts that if the antecedent is true, then the consequent must also be true.

Converse

The converse of a conditional statement is formed by switching the positions of the antecedent and the consequent. This means that for an 'if P then Q' statement, the converse is 'if Q then P'. However, the truth of the original statement does not guarantee the truth of its converse.

Inverse

The inverse of a conditional statement is created by negating both the antecedent and the consequent of the statement. In an 'if P then Q' statement, the inverse is 'if not P then not Q'. The logical equivalence of the inverse to the original statement is not generally assumed.

Contrapositive

The contrapositive of a conditional statement is formed by both swapping and negating the antecedent and the consequent. For an 'if P then Q' statement, the contrapositive is 'if not Q then not P'. This form is logically equivalent to the original 'if-then' statement, meaning it shares the same truth value.

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