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Chapter 3: Problem 41

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

Short Answer

Expert verified
The truth table for the given statement, \(\sim(p \vee q) \wedge \sim r\), is as follows: \n\n\(p\), \(q\), \(r\), \(\sim(p \vee q)\), \(\sim r\), \(\sim(p \vee q) \wedge \sim r\)\n\nT, T, T, F, F, F\n\nT, T, F, F, T, F\n\nT, F, T, F, F, F\n\nT, F, F, F, T, F\n\nF, T, T, F, F, F\n\nF, T, F, F, T, F\n\nF, F, T, T, F, F\n\nF, F, F, T, T, T

Step by step solution

01

Identify all the variables involved

The logical expression contains three variables: \(p\), \(q\), and \(r\). These variables can take on the truth values true (T) or false (F). So, a truth table for the given statement will have \(2^3 = 8\) rows, each representing a unique combination of truth values for \(p\), \(q\), and \(r\).
02

Write down all combinations of truth values

Start by writing the 8 possible combinations of truth values for the variables \(p\), \(q\), and \(r\). They are: TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.
03

Evaluate \(\sim(p \vee q)\)

Next, evaluate \(\sim(p \vee q)\) for each combination of truth values. Here, \(\sim\) denotes NOT, and \(\vee\) denotes OR. So, \(\sim(p \vee q)\) is true only when both \(p\) and \(q\) are false.
04

Evaluate \(\sim r\)

Subsequently, evaluate \(\sim r\) for each combination of truth values. Here, \(\sim\) denotes NOT. So, \(\sim r\) is true when \(r\) is false and vice versa.
05

Evaluate \(\sim(p \vee q) \wedge \sim r\)

Finally, calculate the final column, i.e., the truth value of \(\sim(p \vee q) \wedge \sim r\). Here, \(\wedge\) denotes AND. So, \(\sim(p \vee q) \wedge \sim r\) is true when both \(\sim(p \vee q)\) and \(\sim r\) are true. It's false in all other cases.

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Most popular questions from this chapter

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