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Mathematical Reasoning
medium
Consider the following two propositions:
$P_1: \sim( p \rightarrow \sim q )$
$P_2:( p \wedge \sim q ) \wedge((\sim p ) \vee q )$
If the proposition $p \rightarrow((\sim p ) \vee q )$ is evaluated as $FALSE$, then
A
$P_1$ is TRUE and $P_2$ is FALSE
B
$P_1$ is FALSE and $P_2$ is TRUE
C
Both $P_1$ and $P_2$ are FALSE
D
Both $P_1$ and $P_2$are TRUE
(JEE MAIN-2022)
Solution
| $p$ | $q$ | $\sim p$ | $\sim q$ | $\sim p \vee q$ | $p \rightarrow(\sim p \vee q )$ | $( p \rightarrow \sim q )$ | $\sim( p \rightarrow \sim q )$ | $p \wedge \sim q$ | $p_2$ |
| $T$ | $T$ | $F$ | $F$ | $T$ | $T$ | $F$ | $T$ | $F$ | $F$ |
| $T$ | $F$ | $F$ | $T$ | $F$ | $F$ | $T$ | $F$ | $T$ | $F$ |
| $F$ | $T$ | $T$ |
$F$ |
$T$ | $T$ | $T$ | $F$ | $F$ | $F$ |
| $F$ | $F$ | $T$ | $T$ | $T$ | $T$ | $T$ | $F$ | $F$ | $F$ |
$p \rightarrow(\sim p \vee q )$ is $F$ when $p$ is true $q$ is false
From table
$P_1$ and $P_2$ both are false
Standard 11
Mathematics
