Question

Let $a : \sim(p \wedge \sim r) \vee (\sim q \vee s)$ and $b : (p \vee s) \leftrightarrow (q \wedge r)$. If the truth values of $p$ and $q$ are true and that of $r$ and $s$ are false, then the truth values of $a$ and $b$ are respectively

• A. T, F
• B. T, T
• C. F, F
• D. F, T

Hint:

To speed up evaluation, look for shortcuts: in statement $a$, the second block $(\sim q \vee s) \equiv (F \vee F) \equiv F$. This means the final value of $a$ depends entirely on the first block. Evaluating $\sim(T \wedge T) \equiv F$ gives $F \vee F = F$, which instantly eliminates options (A) and (B)!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given two compound logical statements, $a$ and $b$, constructed with basic connectives. We need to evaluate the final numerical truth value ($T$ or $F$) for both statements using the given individual values: $p \equiv T$, $q \equiv T$, $r \equiv F$, and $s \equiv F$.

Step 2: Key Formula or Approach:
We systematically substitute the truth values into each compound statement and evaluate them step-by-step using standard truth tables for conjunction ($\wedge$), disjunction ($\vee$), negation ($\sim$), and biconditional ($\leftrightarrow$).

Step 3: Detailed Explanation:
Let's first evaluate statement $a$: $$a : \sim(p \wedge \sim r) \vee (\sim q \vee s)$$ Substitute $p \equiv T$, $q \equiv T$, $r \equiv F$, $s \equiv F$: $$a \equiv \sim(T \wedge \sim F) \vee (\sim T \vee F)$$ Since $\sim F \equiv T$ and $\sim T \equiv F$: $$a \equiv \sim(T \wedge T) \vee (F \vee F)$$ Since $T \wedge T \equiv T$ and $F \vee F \equiv F$: $$a \equiv \sim T \vee F$$ $$a \equiv F \vee F \equiv F$$ Now, let's evaluate statement $b$: $$b : (p \vee s) \leftrightarrow (q \wedge r)$$ Substitute the same truth values: $$b \equiv (T \vee F) \leftrightarrow (T \wedge F)$$ Since $T \vee F \equiv T$ and $T \wedge F \equiv F$: $$b \equiv T \leftrightarrow F$$ A biconditional statement ($\leftrightarrow$) is only true when both components have the same truth value. Since they are different here: $$b \equiv F$$ Both statements evaluate to $F$, which gives the pair (F, F).

Step 4: Final Answer:
The truth values of $a$ and $b$ are respectively F, F, which corresponds to option (C).