Question

Use truth tables to show that the following statements are logically equivalent. \(P \Rightarrow Q=(\sim P) \vee Q\)

Short answer

Using a truth table, you can confirm that \(P \Rightarrow Q\) and \((\sim P) \vee Q\) are indeed logically equivalent as they have same truth value under all possible combinations of P and Q.

Step-by-step solution

Definition of Terms

Understanding the symbols used is important. In boolean logic: \(P \Rightarrow Q\) represents 'if P, then Q'; \(\sim P\) represents 'not P'; and \( \vee \) stands for 'OR'.

Constructing the Truth Table

The next step is to construct a truth table that includes all possible combinations of truth values for P and Q. This will give us four combinations: both P and Q can be either true or false.

Calculating Implication Values

Implication (\(P \Rightarrow Q\)) is false only if the antecedent (P) is true and the consequent (Q) is false otherwise it is true. You will fill in the corresponding values of true or false for each combination of P and Q.

Calculating OR Values

For each combination of P and Q, we also want to calculate the value of the right side of our equivalence (\(\sim P \vee Q\)). The 'OR' operation (\( \vee \)) is true if either \(\sim P\) is true, or Q is true. The negation (\(\sim\)) switches the value of P.

Compare the Results

Now, for each combination of P and Q, you compare the results of \(P \Rightarrow Q\) and \((\sim P) \vee Q\). If they are the same for every combination, then the two statements are logically equivalent.