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Chapter 3: Problem 45

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. I'm leaving, and Tom is relieved or Sue is relieved. b. I'm leaving, and it is false that Tom and Sue are not relieved. c. If I'm leaving, then Tom is relieved or Sue is relieved.

Short Answer

Expert verified
All the three given statements are equivalent.

Step by step solution

01

Understanding and Breaking Down Statements

First, identify key events in each statement and how they are logically connected. The key events are - 'I'm leaving', 'Tom is relieved', 'Sue is relieved. The logical operators are 'and', 'or', and 'not'. It is also seen that the negation operator is present in statement b.
02

Applying De Morgan's Laws

De Morgan's laws can be used to simplify complex logical expressions. According to these laws, the negation of a disjunction is the conjunction of the negations, and the negation of a conjunction is the disjunction of the negations. Statement b is simplified using De Morgan's laws: 'It is false that Tom and Sue are not relieved' translates to 'Either Tom is relieved or Sue is relieved', since the negativity is cancelled out by De Morgan's law. Therefore, statement b becomes 'I'm leaving and Either Tom is relieved or Sue is relieved'.
03

Comparing the Statements

On comparing all three statements, it is observed that all statements imply the same meaning. In all statements, the event of 'I'm leaving' results in the relief of either Tom or Sue or both. Hence, all three statements are equivalent.

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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 all people obey the law, then no jails are needed. Some jails are needed. \(\therefore\) Some people do not obey the law.

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'm at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I'm not at the beach, then I don't feel refreshed.

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 it rains or snows, then I read. I am not reading. \(\therefore\) It is neither raining nor snowing.

If you are given an argument in words that contains two premises and a conclusion, describe how to determine if the argument is valid or invalid.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can't use Euler diagrams to determine the validity of an argument if one of the premises is false.
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