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Chapter 3: Problem 44

Determine which, if any, of the three given statements are equivalent. You may use information about a conditional statement's converse, inverse, or contrapositive, De Morgan's laws, or truth tables. a. If you do not file or provide fraudulent information, you will be prosecuted. b. If you file and do not provide fraudulent information, you will not be prosecuted. c. If you are not prosecuted, you filed or did not provide fraudulent information.

Short Answer

Expert verified
Statements a and c are equivalent.

Step by step solution

01

Identify the logical structure of each statement

Translate each statement into a logical form. Consider each component of the statement: a. If not p (you do not file) or q (provide fraudulent information), then r (you will be prosecuted)b. If p (you file) and not q (do not provide fraudulent information), then not r (you will not be prosecuted)c. If not r (you are not prosecuted), then p (you filed) or not q (did not provide fraudulent information)
02

Apply logic operations to identify possible equivalences

Consider the inverse, converse, and contrapositive of each statement. Remember, two statements are logically equivalent if they produce the same truth values.a's contrapositive: If not r (not prosecuted) then p (you filed) or not q (did not provide fraudulent information). b's contrapositive: if r (you will be prosecuted), then not p (you do not file) or q (you provided fraudulent information).c's contrapositive: If not p (you do not file) and q (you provided fraudulent information), then r (you will be prosecuted).
03

Compare the logical forms to determine equivalences

The statement a and c are equivalent as they have the same contrapositive. This means in all situations where a would be true, c would also be true, and vice versa.

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