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Chapter 3: Problem 43

Describe how to obtain the contrapositive of a conditional statement.

Short Answer

Expert verified
The contrapositive of a conditional statement 'If P, then Q.' is 'If not Q, then not P.' So, the contrapositive is formulated by inverting and negating both the hypothesis and conclusion of the original conditional statement.

Step by step solution

01

Understand a Conditional Statement

Firstly, understand what a conditional statement is. A conditional statement, in logic, is a statement of the form 'If P, then Q.', where 'P' is a hypothesis and 'Q' is a conclusion. The condition, 'P', is a proposition that implies the second proposition, 'Q', based on a logical reasoning.
02

Realize the Contrapositive

Once the conditional statement is comprehended, progress towards understanding its contrapositive. The contrapositive of a conditional statement 'If P, then Q.' is 'If not Q, then not P.' This means that if the conclusion 'Q' is false then the condition 'P' is also false.
03

Practice Example

Now to demonstrate these concepts, consider an example. Let's take a conditional statement, 'If a number is 2, then it is even.' The contrapositive of this statement would be, 'If a number is not even, then it is not 2.' This illustrates how to obtain the contrapositive of a conditional statement.

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Most popular questions from this chapter

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If we are to have peace, we must not encourage the competitive spirit. If we are to make progress, we must encourage the competitive spirit. \(\therefore\) We do not have peace and we do not make progress.

Determine whether each argument is valid or invalid. Some natural numbers are even, all natural numbers are whole numbers, and all whole numbers are integers. Thus, some integers are even.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If he was disloyal, his dismissal was justified. If he was loyal, his dismissial was justified. \(\therefore\) His dismissal was justified.

Use Euler diagrams to determine whether each argument is valid or invalid. All dogs have fleas. Some dogs have rabies. Therefore, all dogs with rabies have fleas.

Use Euler diagrams to determine whether each argument is valid or invalid. All actors are artists. Sean Penn is an artist. Therefore, Sean Penn is an actor.
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