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Chapter 3: Problem 12

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

Short Answer

Expert verified
The truth table is as follows: when \(r, p, q\) are all true, \(r \rightarrow(p \vee q)\) is true; when \(r, p\) are true and \(q\) is false, \(r \rightarrow(p \vee q)\) is true; when \(r, q\) are true and \(p\) is false, \(r \rightarrow(p \vee q)\) is true; when only \(r\) is true, \(r \rightarrow(p \vee q)\) is false; when \(p\) and \(q\) are true, \(r \rightarrow(p \vee q)\) is true; in all other cases, \(r \rightarrow(p \vee q)\) is true.

Step by step solution

01

Identify all distinct logical variables

In the given statement, the distinct logical variables or propositions are \(r\), \(p\), and \(q\).
02

Enumerate the Possible Combinations of Truth Values

Given that there are three logical variables, we have \(2^3 = 8\) possible combinations of truth values. Write this down in a table.
03

Evaluate the subexpression

Evaluate the subexpression (\(p \vee q\)) first for each combination of truth values of \(p\) and \(q\) according to the rule of logical disjunction (\(\vee\)). This rule states that 'p OR q' is true iff at least one of \(p\) or \(q\) is true. Write down these results in a new column of the table.
04

Evaluate the Full Statement

Next, evaluate the full logical expression \(r \rightarrow(p \vee q)\) for each combination of truth values. The truth rule states that 'if r, then (p OR q)' is true iff either \(r\) is false, or \(r\) is true and \(p \vee q\) is also true. Record these results in the truth table.

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