2023
JEE Mains
Numerical
JEE Main 2023 (Online) 11th April Morning Shift
The number of ordered triplets of the truth values of $p, q$ and $r$ such that the truth value of the statement $(p \vee q) \wedge(p \vee r) \Rightarrow(q \vee r)$ is True, is equal to ___________.
Correct Answer: 7
Explanation:
$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline \boldsymbol{p} & \boldsymbol{q} & \boldsymbol{r} & \boldsymbol{p} \vee \boldsymbol{q} & \boldsymbol{p} \vee \boldsymbol{r} & \begin{array}{c}
(\boldsymbol{p} \vee \boldsymbol{q}) \wedge \\
(\boldsymbol{p} \vee \boldsymbol{r})
\end{array} & \boldsymbol{q} \vee \boldsymbol{r} & \begin{array}{c}
(\boldsymbol{p} \vee \boldsymbol{q}) \wedge(\boldsymbol{p} \vee \boldsymbol{r}) \\
\Rightarrow(\boldsymbol{q} \vee \boldsymbol{r})
\end{array} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline
\end{array}
$
Hence, the total number of ordered triplets are 7.
Hence, the total number of ordered triplets are 7.
2022
JEE Mains
Numerical
JEE Main 2022 (Online) 28th June Evening Shift
The maximum number of compound propositions, out of p$\vee$r$\vee$s, p$\vee$r$\vee$$\sim$s, p$\vee$$\sim$q$\vee$s, $\sim$p$\vee$$\sim$r$\vee$s, $\sim$p$\vee$$\sim$r$\vee$$\sim$s, $\sim$p$\vee$q$\vee$$\sim$s, q$\vee$r$\vee$$\sim$s, q$\vee$$\sim$r$\vee$$\sim$s, $\sim$p$\vee$$\sim$q$\vee$$\sim$s that can be made simultaneously true by an assignment of the truth values to p, q, r and s, is equal to __________.
Correct Answer: 9
Explanation:
There are total 9 compound propositions, out of which 6 contain $\sim s$. So if we assign $s$ as false, these 6 propositions will be true.
In remaining 3 compound propositions, two contain $p$ and the third contains $\sim r$. So if we assign $p$ and $r$ as true and false respectively, these 3 propositions will also be true.
Hence maximum number of propositions that can be true are 9.
In remaining 3 compound propositions, two contain $p$ and the third contains $\sim r$. So if we assign $p$ and $r$ as true and false respectively, these 3 propositions will also be true.
Hence maximum number of propositions that can be true are 9.