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Chapter 3: Problem 32

Express each statement in "if ... then" form. (More than one correct wording in "if... then" form may be possible.) Then write the statement's converse, inverse, and contrapositive. All senators are politicians.

Short Answer

Expert verified
The 'If...then' form is: 'If someone is a senator, then they are a politician.' The converse is: 'If someone is a politician, then they are a senator.' The inverse is: 'If someone is not a senator, then they are not a politician.' The contrapositive is: 'If someone is not a politician, then they are not a senator.'

Step by step solution

01

Rewrite in 'If...then' form

First, we recast the statement 'All senators are politicians' in 'If...then' form. The statement can be rewritten as: 'If someone is a senator, then they are a politician.'
02

Determine and write down the converse

The converse of a statement swaps the hypothesis and the conclusion. Thus, the converse of our 'If...then' statement 'If someone is a senator, then they are a politician' becomes: 'If someone is a politician, then they are a senator.'
03

Determine and write down the inverse

The inverse of a statement negates both the hypothesis and the conclusion. The inverse of 'If someone is a senator, then they are a politician' is: 'If someone is not a senator, then they are not a politician.'
04

Determine and write down the contrapositive

The contrapositive of a statement swaps the hypothesis and conclusion and then negates them. Thus, the contrapositive for 'If someone is a senator, then they are a politician' is: 'If someone is not a politician, then they are not a senator.'

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Most popular questions from this chapter

Describe what is meant by a valid argument.

Use Euler diagrams to determine whether each argument is valid or invalid. All multiples of 6 are multiples of 3 . Eight is not a multiple of \(6 .\) Therefore, 8 is not a multiple of 3 .

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 I am tired or hungry, I cannot concentrate. I can concentrate. \(\therefore\) I am neither tired nor hungry.

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 I am at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I am at the beach, then I feel refreshed.

Determine whether each argument is valid or invalid. No \(A\) are \(B\), some \(A\) are \(C\), and all \(C\) are \(D\). Thus, some \(D\) are \(B\)
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