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Chapter 3: Problem 27

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

Short Answer

Expert verified
The truth table for \(\sim(\sim p \vee q)\) is as follows: \(p=(T, T, F, F)\), \(q=(T, F, T, F)\), \(\sim p =(F, F, T, T)\), \(\sim p \vee q=(T, F, T, T)\), and \(\sim(\sim p \vee q)=(F, T, F, F)\).

Step by step solution

01

Set up the Truth Table

Begin by setting up the truth table with columns representing \(p\), \(q\), \(\sim p\), \(\sim p \vee q\), and \(\sim(\sim p \vee q)\). The first two columns for \(p\) and \(q\) represent all possible combinations of truth values for the propositional variables: \((T, T)\), \((T, F)\), \((F, T)\), \((F, F)\).
02

Fill in the Values for \(\sim p\)

Fill in the truth values for the \(\sim p\) column. This represents the negation of \(p\), i.e., it is true when \(p\) is false and false when \(p\) is true.
03

Fill in the Values for \(\sim p \vee q\)

Fill in the truth values for the \(\sim p \vee q\) column. This represents the disjunction of \(\sim p\) and \(q\), i.e., it is true when either \(\sim p\) or \(q\) (or both) is true, and false otherwise.
04

Fill in the Values for \(\sim(\sim p \vee q)\)

Fill in the truth values for the \(\sim(\sim p \vee q)\) column. This represents the negation of the \(\sim p \vee q\) column, i.e., it is true when \(\sim p \vee q\) is false, and false when \(\sim p \vee q\) is true.

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Most popular questions from this chapter

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

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