Question

Let \( p : 57 \) is an odd prime number,
\( q : 4 \) is a divisor of 12,
\( r : 15 \) is the LCM of 3 and 5 be three simple logical statements.
Which one of the following is true?

• A. $p \lor (\sim q \land r)$
• B. $\sim p \lor (q \land r)$
• C. $(p \land q) \lor \sim r$
• D. $(p \lor q) \land \sim r$
• E. $\sim p \land (\sim q \land r)$

Hint:

In mathematical logic problems, the hardest part is often checking the "math facts" in the statements (like checking if 57 is prime). Once the T/F values are set, the logic table follows standard rules.

Answer 1

The Correct Option is B

Concept: To determine the truth value of compound logical statements, we must first establish the truth value (T or F) of the individual component statements $p$, $q$, and $r$.
• $\lor$ (OR): True if at least one component is True.
• $\land$ (AND): True only if both components are True.
• $\sim$ (NOT): Reverses the truth value.

Step 1: Determine the truth values of $p, q, r$.

• $p$: "57 is an odd prime." $57 = 3 \times 19$, so it is not prime. Value: F.
• $q$: "4 is a divisor of 12." $12 \div 4 = 3$. Value: T.
• $r$: "15 is the LCM of 3 and 5." 3 and 5 are coprime, so $LCM = 3 \times 5 = 15$. Value: T.

Step 2: Test the options using $p=F, q=T, r=T$.
(A) $F \lor (\sim T \land T) \Rightarrow F \lor (F \land T) \Rightarrow F \lor F = F$.
(B) $\sim F \lor (T \land T) \Rightarrow T \lor T = T$.
(C) $(F \land T) \lor \sim T \Rightarrow F \lor F = F$.
(D) $(F \lor T) \land \sim T \Rightarrow T \land F = F$.

Step 3: Conclusion.
Option (B) evaluates to True.