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Chapter 3: Problem 28

Write the converse, inverse, and contrapositive of each statement. If all hard workers are successful, then some people are not hard workers.

Short Answer

Expert verified
The converse of the statement is 'If some people are not hard workers, then all hard workers are successful'. The inverse is 'If not all hard workers are successful, then some people are hard workers'. The contrapositive is 'If not some people are not hard workers, then not all hard workers are successful'.

Step by step solution

01

Identify the Hypothesis and Conclusion

In the statement 'If all hard workers are successful, then some people are not hard workers.', identify the hypothesis (p) and the conclusion (q). Here, p represents the statement 'all hard workers are successful', and q represents the statement 'some people are not hard workers'.
02

Formulate the Converse

The converse is formed by reversing the hypothesis and the conclusion. So, the converse of the original statement is 'If some people are not hard workers, then all hard workers are successful'.
03

Formulate the Inverse

The inverse is formed by negating both the hypothesis and the conclusion. So the inverse of the statement is 'If not all hard workers are successful, then some people are hard workers'.
04

Formulate the Contrapositive

The contrapositive entails both switching and negating the hypothesis and the conclusion. So the contrapositive of the statement will be 'If not some people are not hard workers, then not all hard workers are successful'.

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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 I am tired or hungry, I cannot concentrate. I cannot concentrate. \(\therefore\) I am tired or hungry.

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