Question

If truth values of statements p, q are true, and r, s are false, then the truth values of the following statement patterns are respectively:
$a:\sim(p\wedge\sim r)\vee(\sim q\vee s)$
$b:(\sim q\wedge\sim r)\leftrightarrow(p\vee s)$
$c:(\sim p\vee q)\rightarrow(r\wedge\sim s)$

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

Hint:

Logic Tip: Remember that the implication $P \rightarrow Q$ is only False when $P$ is True and $Q$ is False (the broken promise). For biconditional $P \leftrightarrow Q$, it is True only when both components share the identical truth value (both True or both False).

Answer 1

The Correct Option is B

Concept:
In mathematical logic, evaluate compound statements by substituting the given truth values and applying the standard truth tables for logical connectives: Conjunction ($\wedge$, AND), Disjunction ($\vee$, OR), Negation ($\sim$, NOT), Implication ($\rightarrow$, IF-THEN), and Biconditional ($\leftrightarrow$, IF AND ONLY IF).
Step 1: List the given truth values.
Given: $p \equiv T$ $q \equiv T$ $r \equiv F$ $s \equiv F$
Step 2: Evaluate statement pattern (a).
$a: \sim(p \wedge \sim r) \vee (\sim q \vee s)$ Substitute the values: $\equiv \sim(T \wedge \sim F) \vee (\sim T \vee F)$ $\equiv \sim(T \wedge T) \vee (F \vee F)$ $\equiv \sim(T) \vee (F)$ $\equiv F \vee F$ $\equiv F$
Step 3: Evaluate statement pattern (b).
$b: (\sim q \wedge \sim r) \leftrightarrow (p \vee s)$ Substitute the values: $\equiv (\sim T \wedge \sim F) \leftrightarrow (T \vee F)$ $\equiv (F \wedge T) \leftrightarrow (T)$ $\equiv F \leftrightarrow T$ $\equiv F$
Step 4: Evaluate statement pattern (c).
$c: (\sim p \vee q) \rightarrow (r \wedge \sim s)$ Substitute the values: $\equiv (\sim T \vee T) \rightarrow (F \wedge \sim F)$ $\equiv (F \vee T) \rightarrow (F \wedge T)$ $\equiv T \rightarrow F$ $\equiv F$ The truth values are F, F, and F respectively.