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Chapter 3: Problem 28

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

Short Answer

Expert verified
The truth table for the logical expression \(\sim(p \wedge \sim q)\) will have four rows for the four possible combinations of \(p\) and \(q\). For each combination, the value of the logical expression is calculated using the defined operations: NOT, AND, and OR. The final truth values can be found in the column under \(\sim(p \wedge \sim q)\). It's crucial to calculate the inner operations first before the outer ones following the rules of the Boolean Algebra.

Step by step solution

01

Define Boolean Variables

Begin by defining the possible values for the boolean variables \(p\) and \(q\). Boolean variables can either be True (T) or False (F). Therefore, there are \(2^2=4\) possibilities: TT, TF, FT, FF.
02

Evaluate the Inner NOT Operation

Now we calculate the inner NOT operation (\(\sim q\)). This operation changes True to False, and vice versa. Form a column in the table under \(\sim q\) with the negated values of \(q\). If \(q\) is True, \(\sim q\) is False. If \(q\) is False, \(\sim q\) is True.
03

Evaluate the AND Operation

Next, calculate the AND operation (\(p \wedge \sim q\)). This operation is only true when both of its operands are True. Form a new column in the table under \(p \wedge \sim q\) with its results.
04

Evaluate the Outer NOT Operation

Finally, calculate the outer NOT operation (\(\sim(p \wedge \sim q)\)). This operation changes True to False and vice versa. Form a new column in the truth table with its results. The values in this column are the final truth values of the given logical expression.

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