1. The problem is to understand and simplify the expression $P \equiv (\sim r^{\sim q})$. 2. Typically, the symbol $\sim$ denotes negation (NOT), and $^$ sometimes represents XOR or exclusive OR. 3. If we interpret $^{\sim q}$ as XOR with $\sim q$, the expression is $P \equiv \sim r \oplus \sim q$. 4. Using the property of XOR, $\sim r \oplus \sim q = r \oplus q$ because XOR with negations flips both inputs. 5. Therefore, the simplified form of $P$ is $P = r \oplus q$. 6. This means $P$ is TRUE when exactly one of $r$ or $q$ is TRUE.