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If $(\sim \mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{r}$ is false then the truth values of $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are respectively
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$\mathbf{F}, \mathbf{T}, \mathbf{F}$
Given $(\sim p \wedge q) \rightarrow r$ is false $T \rightarrow F \equiv F$
We know that $T \rightarrow F=F$
$\therefore \sim p \wedge q=T \quad$ and $r \equiv F$
Also we know that $\mathrm{T} \wedge \mathrm{T}=\mathrm{T}$
$\therefore \mathrm{q} \equiv \mathrm{T}$ and $\sim \mathrm{p}=\mathrm{T} \Rightarrow \mathrm{p}=\mathrm{F}$
We know that $T \rightarrow F=F$
$\therefore \sim p \wedge q=T \quad$ and $r \equiv F$
Also we know that $\mathrm{T} \wedge \mathrm{T}=\mathrm{T}$
$\therefore \mathrm{q} \equiv \mathrm{T}$ and $\sim \mathrm{p}=\mathrm{T} \Rightarrow \mathrm{p}=\mathrm{F}$
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