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Chapter 3: Problem 36

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

Short Answer

Expert verified
The resulting truth table looks like this:| p | q | \(\sim q\) | \(p \vee \sim q\) | \(p \vee q\) | \((p \vee \sim q) \wedge(p \vee q)\) ||:-:|:-:|:-:|:-:|:-:|:-:|| T | T | F | T | T | T || T | F | T | T | T | T || F | T | F | F | T | F || F | F | T | T | F | F |So, for the given statement, when 'p' is True, the statement is always True, and otherwise, the truth value of the statement is the same with the truth value of 'q'.

Step by step solution

01

Identify the Variables

In this statement, 'p' and 'q' are the variables. These variables can either be True (T) or False (F).
02

Construct the Table with All Possible States for The Variables

There are 4 possible outcomes for two variables, they are: TT, TF, FT, FF.
03

Write Down the Truth Values for \( \sim q \)

We flip the truth value of 'q'. So, if q is T, \( \sim q \) is F and if q is F, \( \sim q \) is T.
04

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

We evaluate the OR operation on 'p' and \( \sim q \). In OR operation, the result is True if either of the operand is True.
05

Evaluate \( (p \vee q) \)

We evaluate the OR operation on 'p' and 'q'. Similar to step 4, the result is True if either 'p' or 'q' is True.
06

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

We then evaluate the AND operation on the outcome of step 4 and step 5. In AND operation, the result is only true when both operands are true. The final row of the table provides the truth value of the entire statement for each combination of 'p' and 'q'.

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