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Chapter 3: Problem 2

a. Use a truth table to show that \(p \rightarrow q\) and \(\sim p \vee q\) are equivalent. b. Use the result from part (a) to write a statement that is equivalent to If a number is even, then it is divisible by \(2 .\)

Short Answer

Expert verified
a. The logical expressions \(p \rightarrow q\) and \(\sim p \vee q\) are equivalent. b. An equivalent statement to 'If a number is even, then it is divisible by 2' is 'Either the number is not even or it is divisible by 2'.

Step by step solution

01

Establish the truth table for logical expression \(p \rightarrow q\)

First, construct a truth table with three columns: \(p\), \(q\) and \(p \rightarrow q\). The first two columns will be filled with all possible combinations of truth values of \(p\) and \(q\) (that is, TT, TF, FT, FF). In the third column, determine the value of \(p \rightarrow q\) for each combination of \(p\) and \(q\) according to the definition of implication (if \(p\) is true and \(q\) is false, then \(p \rightarrow q\) is false; otherwise, \(p \rightarrow q\) is true).
02

Establish the truth table for logical expression \(\sim p \vee q\)

Then, create a truth table with three columns: \(p\), \(q\) and \(\sim p \vee q\). Fill the first two columns with all possible combinations of truth values of \(p\) and \(q\) again. In the third column, determine the value of \(\sim p \vee q\) for each combination of \(p\) and \(q\) according to the definition of negation and disjunction (if \(p\) is true, then \(\sim p\) is false, and vice versa; \(\sim p \vee q\) is true if either or both of \(\sim p\) and \(q\) are true, and false otherwise).
03

Compare the truth tables

Compare the third column of the first truth table and the third column of the second truth table. If they are identical, it means the two logical expressions \(p \rightarrow q\) and \(\sim p \vee q\) are logically equivalent, as required.
04

Interpret the logical equivalence

In light of the equivalence between \(p \rightarrow q\) and \(\sim p \vee q\), interpret 'If a number is even (denoted by \(p\)), then it is divisible by 2 (denoted by \(q\))' as 'either the number is not even (\(\sim p\)) or it is divisible by 2 (\(q\))', which is an equivalent statement.

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Most popular questions from this chapter

Use Euler diagrams to determine whether each argument is valid or invalid. All humans are warm-blooded. No reptiles are warm-blooded. Therefore, no reptiles are human.

In Exercises 1-24, use Euler diagrams to determine whether each argument is valid or invalid. All writers appreciate language. All poets are writers. Therefore, all poets appreciate language.

Conservative commentator Rush Limbaugh directed this passage at liberals and the way they think about crime. Of course, liberals will argue that these actions [contemporary youth crime] can be laid at the foot of socioeconomic inequities, or poverty. However, the Great Depression caused a level of poverty unknown to exist in America today, and yet I have been unable to find any accounts of crime waves sweeping our large cities. Let the liberals chew on that. (See, I Told You So, p. 83) Limbaugh's passage can be expressed in the form of an argument: If poverty causes crime, then crime waves would have swept American cities during the Great Depression. Crime waves did not sweep American cities during the Great Depression. \(\therefore\) Poverty does not cause crime. (Liberals are wrong.) Translate this argument into symbolic form and determine whether it is valid or invalid.

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &p \rightarrow q \\ &\underline{q \rightarrow r} \\ &\therefore \sim p \rightarrow \sim r \end{aligned} $$

Write an original argument in words for the direct reasoning form.
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