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Worksheet Truth Tables and Deductions

A truth table consists of rows for every possible combination of truth values for the propositional variables we are working with, and a column for each statement we work with. Each column then contains the truth values corresponding the the choices of truth values for the propositional variables in that row.

1.

What can you conclude about the truth values of the statements \(P\text{,}\) \(Q\text{,}\) and \(R\) if you know that the statement \((P \imp Q) \wedge R\) is true? Complete the truth table below to find out.
\(P\) \(Q\) \(R\) \(P \imp Q\) \((P \imp Q) \wedge R\)
\(T\) \(T\) \(T\)
\(T\) \(T\) \(F\)
\(T\) \(F\) \(T\)
\(T\) \(F\) \(F\)
\(F\) \(T\) \(T\)
\(F\) \(T\) \(F\)
\(F\) \(F\) \(T\)
\(F\) \(F\) \(F\)

2.

Make a truth table for the statements \(P \imp Q\) and \(\neg Q \imp \neg P\text{.}\) What do you notice?

3.

Make a truth table for the statements \(\neg(P \wedge Q)\) and \(\neg P \vee \neg Q\text{.}\) What do you notice?

4.

Write a truth table that includes the following statements: \(Q \imp (P \vee R)\text{,}\) \(Q \vee R\text{,}\) and \(P \vee R\text{.}\) Is there a relationship you can identify between the three statements?

5.

Verify that the following argument is valid. That is, show it is a valid deduction rule.
\(R \imp S\)
\(\neg S\)
\(Q \vee R\)
\(\therefore\) \(Q\)
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