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Chapter 3: Problem 7

Construct a truth table for the given statement. \((p \rightarrow q) \wedge \sim q\)

Short Answer

Expert verified
The truth table for \( (p \rightarrow q) \wedge \sim q \) will be as follows: \[\begin{align*}p && q && p \rightarrow q && \sim q && (p \rightarrow q) \wedge \sim q \T && T && T && F && F \T && F && F && T && F \F && T && T && F && F \F && F && T && T && T \\end{align*}\]

Step by step solution

01

Define the Variables

The most basic parts of this statement are the propositions p and q. They can each be true (T) or false (F). Start by creating a table with columns for p and q, and list all possible combinations of truth values.
02

Evaluate the Implication

Remember the truth values for an implication statement \( p \rightarrow q \) are: TT-T, TF-F, FT-T, FF-T. Create a column for the implication \( p \rightarrow q \), and fill in its truth values based on the combinations of p and q in your table.
03

Evaluate the Negation

Next, evaluate the negation \( \sim q \). The negation of a true statement is false and the negation of a false statement is true. Create another column for \( \sim q \), and fill in its truth values based on the values of q in your table.
04

Evaluate the Conjunction

The final step is to evaluate the conjunction: \( (p \rightarrow q) \wedge \sim q \). The truth value of the conjunction is true only if both of its components are true. Create a final column for the entire statement, and fill in its truth values based on the values of \( p \rightarrow q \) and \( \sim q \) in your table.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made Euler diagrams for the premises of an argument and one of my possible diagr芒ms did not illustraate the conclusion, so the argument is invalid.

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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.
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