Animated Solution for Mathematics - Sets and Relations: If $q$ is false and $p\wedge q\leftrightarrow r$ is true, then which one of the following statements is a tautology?

We are given two fundamental logical constraints. First, we know that $q$ is false, denoted as $q=F$.

Second, we are given that the biconditional statement $(p\wedge q)\leftrightarrow r$ is true. These two pieces of information serve as our anchor points for the entire logical derivation.

We begin by substituting the known value $q=F$ into the expression $(p\wedge q)\leftrightarrow r\equiv T$.

Applying the identity law of logic, we know that any statement $p$ conjoined with false results in false:

Consequently, the expression simplifies to:

The biconditional operator $\leftrightarrow$ requires both sides to have the same truth value to result in true. Since the left side is $F$, the right side $r$ must also be $F$ for the entire statement to hold true.

Thus, we have determined that $r=F$.

We now evaluate the candidate expression $(p\wedge r)\to (p\vee r)$ to determine if it is a tautology. Substituting our derived value $r=F$, the expression becomes:

We simplify the components using basic logical identities. On the left side, $(p\wedge F)$ simplifies to $F$. On the right side, $(p\vee F)$ simplifies to $p$.

The expression now reduces to:

In formal logic, an implication $A\to B$ is only false when the hypothesis $A$ is true and the conclusion $B$ is false. If the hypothesis $A$ is false, the implication is automatically true, regardless of the truth value of $B$.

This phenomenon is known as a vacuous truth. Since our hypothesis is $F$, the statement $F\to p$ is true for any value of $p$.

Therefore, the expression $(p\wedge r)\to (p\vee r)$ is a tautology.