Negation of inverse of the following statement pattern $(p \wedge q) \rightarrow(p \vee \sim q)$ is
Negation of inverse of the following statement pattern $(p \wedge q) \rightarrow(p \vee \sim q)$ is
- $\mathrm{p}$
- $\sim q$
- $\sim p$
- $\mathrm{q}$
Solution
Inverse of $(p \wedge q) \rightarrow(p \vee \sim q)$ is
$\begin{aligned}
& \sim(p \wedge q) \rightarrow \sim(p \vee \sim q) \\
& \equiv \sim[\sim(p \wedge q)] \vee \sim(p \vee \sim q) \ldots[p \rightarrow q \equiv \sim p \vee q]
\end{aligned}$
$\begin{aligned}
& \equiv(p \wedge q) \vee(\sim p \wedge q)...[De Morgan's law]\\
& \equiv(q \wedge p) \vee(q \wedge \sim p)...[Commutative law] \\
& \equiv q \wedge(p \vee \sim p)...[Distributive law] \\
& \equiv q \wedge T...[Complement law] \\
& \equiv q...[Identity law]
\end{aligned}$
$\therefore \quad$ Negation of inverse of $(p \wedge q) \rightarrow(p \vee \sim q)$ is $\sim q$
Asked in: MHT CET 2023 (13 May Shift 1)
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