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13 November, 18:36

What are the converse, inverse, and contrapositive of the following true conditional? What are the truth values of each?

If a statement is false, give a counterexample.

If a figure is a rectangle, then it is a parallelogram.

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  1. 13 November, 19:24
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    In logic, the converse of a conditional statement is the result of reversing its two parts. For example, the statement P → Q, has the converse of Q → P.

    For the given statement, 'If a figure is a rectangle, then it is a parallelogram.' the converse is 'if a figure is a parallelogram, then it is rectangle.'

    As can be seen, the converse statement is not true, hence the truth value of the converse statement is false.

    The inverse of a conditional statement is the result of negating both the hypothesis and conclusion of the conditional statement. For example, the inverse of P → Q is ~P → ~Q.

    For the given statement, 'If a figure is a rectangle, then it is a parallelogram.' the inverse is 'if a figure is not a rectangle, then it is not a parallelogram.'

    As can be seen, the inverse statement is not true, hence the truth value of the inverse statement is false.

    The contrapositive of a conditional statement is switching the hypothesis and conclusion of the conditional statement and negating both. For example, the contrapositive of P → Q is ~Q → ~P.

    For the given statement, 'If a figure is a rectangle, then it is a parallelogram.' the contrapositive is 'if a figure is not a parallelogram, then it is not a rectangle.'

    As can be seen, the contrapositive statement is true, hence the truth value of the contrapositive statement is true.
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