Question

Suppose \(P\) is false and that the statement \((R \Rightarrow S) \Leftrightarrow(P \wedge Q)\) is true. Find the truth values of \(R\) and \(S\). (This can be done without a truth table.)

Short answer

Therefore, the truth values of \(R\) and \(S\) are that \(R\) is true and \(S\) is false.

Step-by-step solution

Understanding the Biconditional

The biconditional operator \(\Leftrightarrow\) means that both sides of the equation are equivalent, so either both are true or both are false. If the statement \((R \Rightarrow S) \Leftrightarrow (P \wedge Q)\) is true, then both \((R \Rightarrow S)\) and \((P \wedge Q)\) either have to be true or have to be false.

Analyzing the Right Side of the Biconditional

Given that \(P\) is known to be false and looking at the right-side of the biconditional \((P \wedge Q)\), it must also be false because a conjunction \(\wedge\) is only true when both arguments are true.

Analyzing the Left Side of the Biconditional

Since the right side of the biconditional is false and the whole biconditional is known to be true, the left-side of the biconditional \((R \Rightarrow S)\) has to be false too. This is because both parts of the biconditional need to be the same. In a implication \(\Rightarrow\), the only way it is false is if \(R\) is true and \(S\) is false.

Determining the Truth Values of R and S

Having determined that the left-side of the biconditional \((R \Rightarrow S)\) must be false, it means that the truth values of \(R\) and \(S\) have to be such that \(R\) is true and \(S\) is false. This is the only case when an implication is false - if the antecedent \(R\) is true and the consequent \(S\) is not.