Question

\((\sim P) \wedge(P \Rightarrow Q)\) and \(\sim(Q \Rightarrow P)\)

Short answer

Both expressions are only true if P is false and Q is true.

Step-by-step solution

Understanding the Logic Operators

First, let's clarify the symbols: \(\sim\) means 'not', \(\wedge\) means 'and', and \(\Rightarrow\) means 'if...then' or 'implies'. The expression \((\sim P) \wedge(P \Rightarrow Q)\) means 'not P and P implies Q', and the expression \(\sim(Q \Rightarrow P)\) means 'not Q implies P'. The problem is to evaluate these two expressions.

Analyzing the First Expression

The expression \((\sim P) \wedge(P \Rightarrow Q)\) can be interpreted in the following way: P is not true and if P is true, then Q would be true. This expression is only true if both P is false and Q is true, otherwise it would be false.

Analyzing the Second Expression

The expression \(\sim(Q \Rightarrow P)\) can be interpreted as follows: It is not true that if Q is true, then P would be true. In the absence of any other information, the only condition for this expression being true would be if Q is true and P is false.

Comparing the Results

Comparing the conditions necessary for both expressions to be true, we see that they would only both be true if P is false and Q is true.