Question

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

Short answer

Yes, the given pairs of statements \(P \wedge Q\) and \(\sim(\sim P \vee \sim Q)\) are logically equivalent.

Step-by-step solution

- Break down the first statement

The first statement is quite straightforward. It is \(P \wedge Q\), which represents that both \(P\) and \(Q\) must be true.

- Break down the second statement

The second statement is \(\sim(\sim P \vee \sim Q)\), which represents the negation of '\(P\) is not true OR \(Q\) is not true'. This can be broken down using De Morgan's law which states that the negation of a disjunction is the conjunction of the negations. This gives us \(P \wedge Q\).

- Compare the two results

After applying De Morgan's law, the second expression simplifies to \(P\wedge Q\), which is the same as the first expression. Hence, both expressions are logically equivalent.