Question

Decide whether or not the following pairs of statements are logically equivalent. \(P \vee(Q \wedge R)\) and \((P \vee Q) \wedge R\)

Short answer

No, the logical expressions \(P \vee (Q \wedge R)\) and \((P \vee Q) \wedge R\) are not equivalent.

Step-by-step solution

Apply the Distributive Law

Firstly, the Distributive Law may be applied to the first expression \(P \vee(Q \wedge R)\) . This results in two terms as follows: \((P \vee Q) \wedge (P \vee R)\)

Compare the Expressions

Comparing \((P \vee Q) \wedge (P \vee R)\) and \((P \vee Q) \wedge R\), it can be seen that they are not equivalent. The expressions would only have been equivalent if they had the same logical connectives involving the same terms.

Conclusion

Since the expressions are not equivalent, the answer is No, the given pairs are not logically equivalent.