Question

If $p$ and $q$ are true and r and s are false statements, then which of the following is true?

• A. $\left(q \wedge r\right)\vee \left(\sim p \wedge s\right)$
• B. $\left(\sim p \rightarrow q\right) \leftrightarrow \left(r \wedge s\right)$
• C. $\left(p \rightarrow q\right) \vee \left(r \leftrightarrow s\right)$
• D. $\left(p \wedge \sim r\right)\wedge \left(\sim q \vee s\right)$

Answer 1

Answer: (C)

We have statements $p, q \rightarrow T$ and $r, s \rightarrow F$
Option $(a) (q \wedge r) \vee(\sim p \wedge s) \equiv(T \wedge F) \vee(F \wedge F)$
$\equiv F \vee F \equiv F$
Option (b) $(\sim p \rightarrow q) \leftrightarrow(r \wedge s) \equiv(p \vee q) \rightarrow(r \vee s)$
$(\because p \rightarrow q \equiv \sim p \vee q)$
$\equiv(\sim p \vee q) \vee(r \wedge s) \wedge((p \vee q) \vee(\sim r \wedge s)))$
$(\because p \leftrightarrow q \equiv(\sim p \vee q) \wedge(p \vee-q))$
$\equiv(\sim(T \vee T) \vee(F \wedge F) \wedge((T \vee F) \vee(\sim(F \wedge F)))$
$\equiv(F \vee F) \wedge(T \vee T) \equiv F \wedge T \equiv F$
Option (c) $(p \rightarrow q) \vee(r \leftrightarrow s)$
$\equiv(\sim p \vee q) \vee((\sim r \vee s) \wedge(r \vee \sim s))(\because p \rightarrow q \equiv \sim p \vee q)$
$\equiv(F \vee T) \vee((T \vee F) \wedge(F \vee T))$
$\equiv T \vee(T \wedge T) \equiv T . \vee T \equiv T$
Option (d) do similar as option (a).