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If $\left( {p \wedge \sim q} \right) \wedge \left( {p \wedge r} \right) \to \sim p \vee q$ is false, then the truth values of $p, q$ and $r$ are respectively
$F, T, F$
$T, F, T$
$F, F, F$
$T, T, T$
Solution
AS the truth table for $\left( { \sim p \wedge \sim q} \right) \wedge \left( {p \wedge r} \right) \to \sim p \wedge q$ is false, then only possible values of $(p,q,r)$ is $(T,F,T,)$
|
$p$ |
$q$ |
$r$ |
${ \sim q}$ |
$p \wedge \sim q$ |
$p \wedge r$ |
$\sim p$ |
$ \sim p \vee q$ |
$\left( {p \wedge \sim q} \right) \wedge \left( {p \wedge r} \right)$ |
$\left( {p \wedge \sim q} \right) \wedge \left( {p \wedge \sim r} \right) \to \sim p \vee q$ |
|
$T$ |
$T$ |
$T$ |
$F$ |
$F$ |
$T$ |
$F$ |
$T$ |
$F$ |
$T$ |
|
$T$ |
$F$ |
$T$ |
$T$ |
$T$ |
$T$ |
$F$ |
$F$ |
$T$ |
$F$ |
|
$T$ |
$T$ |
$F$ |
$F$ |
$F$ |
$F$ |
$F$ |
$T$ |
$F$ |
$T$ |
|
$F$ |
$T$ |
$T$ |
$F$ |
$F$ |
$F$ |
$T$ |
$T$ |
$F$ |
$T$ |
|
$F$ |
$F$ |
$T$ |
$T$ |
$F$ |
$F$ |
$T$ |
$T$ |
$F$ |
$T$ |
|
$F$ |
$T$ |
$F$ |
$F$ |
$F$ |
$F$ |
$T$ |
$T$ |
$F$ |
$T$ |
|
$T$ |
$F$ |
$F$ |
$T$ |
$T$ |
$F$ |
$F$ |
$F$ |
$F$ |
$T$ |
|
$F$ |
$F$ |
$F$ |
$T$ |
$F$ |
$F$ |
$T$ |
$T$ |
$F$ |
$T$ |
