Question

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

Short answer

Yes, \(\sim(P \Rightarrow Q)\) and \(P \wedge \sim Q\) are logically equivalent.

Step-by-step solution

Define the Logical Statements

The first logical statement is \(\sim(P \Rightarrow Q)\), which can be read as 'not (P implies Q)'. The second logical statement is \(P \wedge \sim Q\), which can be read as 'P and not Q'.

Understand the Implication Statement Logic

The implication statement 'P implies Q' is true if and only if P is false or Q is true. So, the negation of an implication \(\sim(P \Rightarrow Q)\) is true when P is true and Q is false.

Find an Equivalent Expression for the First Statement

The statement \(P \wedge \sim Q\) means both 'P is true' and 'Q is false'. Now, comparing this statement to the equivalent one derived for \(\sim(P \Rightarrow Q)\) (P is true and Q is false), it can be seen that both are logically equivalent.