If truth values of statements $p, q$ are true, and $r$, $s$ are false, then the truth values of the…

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 $\mathrm{a}: \sim(\mathrm{p} \wedge \sim \mathrm{r}) \vee(\sim \mathrm{q} \vee \mathrm{s})$ $\mathrm{b}:(\sim \mathrm{q} \wedge \sim \mathrm{r}) \leftrightarrow(\mathrm{p} \vee \mathrm{s})$ $c:(\sim p \vee q) \rightarrow(r \wedge \sim s)$
  1. $T,F,F$
  2. $F,F,F$
  3. $F,T,T$
  4. $T,F,T$

Solution

a. $\begin{aligned} & \sim(\mathrm{p} \wedge \sim \mathrm{r}) \vee(\sim \mathrm{q} \vee \mathrm{s}) \\ & \equiv \sim(\mathrm{T} \wedge \sim \mathrm{F}) \vee(\sim \mathrm{T} \vee \mathrm{F}) \\ & \equiv(\mathrm{F} \vee \mathrm{F}) \vee(\mathrm{F} \vee \mathrm{F}) \\ & \equiv \mathrm{F} \vee \mathrm{F} \\ & \equiv \mathrm{F} \end{aligned}$ b. $\begin{aligned} & (\sim \mathrm{q} \wedge \sim \mathrm{r}) \leftrightarrow(\mathrm{p} \vee \mathrm{s}) \\ & \equiv(\sim \mathrm{T} \wedge \sim \mathrm{F}) \leftrightarrow(\mathrm{T} \vee \mathrm{F}) \\ & \equiv \mathrm{F} \leftrightarrow \mathrm{T} \\ & \equiv \mathrm{F} \end{aligned}$ c. $\begin{aligned} & (\sim p \vee q) \rightarrow(r \wedge \sim s) \\ & \equiv(\sim T \vee T) \rightarrow(F \wedge \sim F) \\ & \equiv(F \vee T) \rightarrow(F \wedge T) \\ & \equiv T \rightarrow F \\ & \equiv F \end{aligned}$

Asked in: MHT CET 2023 (09 May Shift 1)

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