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Mathematical Reasoning
hard
Among the statements:
$(S1)$ $\quad(( p \vee q ) \Rightarrow r ) \Leftrightarrow( p \Rightarrow r )$
$(S2) \quad(( p \vee q ) \Rightarrow r ) \Leftrightarrow(( p \Rightarrow r ) \vee( q \Rightarrow r ))$
A
Only $(S1)$ is a tautology
B
Neither $(S1)$ nor $(S2)$ is a tautology
C
Only $(S2)$ is a tautology
D
Both $(S1)$ and $(S2)$ are tautologies
(JEE MAIN-2023)
Solution
$S _1 \equiv(( p \vee q ) \Rightarrow r ) \Leftrightarrow( p \Rightarrow r )$
| $p$ | $q$ | $r$ | $p \vee q$ | $(p \vee q) \Rightarrow r$ | $p \Rightarrow r$ | $((p \vee q) \Rightarrow r) \Leftrightarrow(p \Rightarrow r)$ |
| $T$ | $T$ | $T$ | $T$ | $T$ | $T$ | $T$ |
| $T$ | $T$ | $F$ | $T$ | $F$ | $F$ | $T$ |
| $T$ | $F$ | $T$ | $T$ | $T$ | $T$ | $T$ |
| $F$ | $T$ | $T$ | $T$ | $T$ | $T$ | $T$ |
| $T$ | $F$ | $F$ | $T$ | $F$ | $F$ | $T$ |
| $F$ | $T$ | $F$ | $T$ | $F$ | $T$ | $T$ |
| $F$ | $F$ | $F$ | $T$ | $T$ | $T$ | $T$ |
$S_2 \equiv(p \vee q) \Rightarrow r \Leftrightarrow((p \Rightarrow r) \vee(q \Rightarrow r))$
| $p$ | $q$ | $r$ | $(p \vee q) \Rightarrow r$ | $p \Rightarrow r$ | $q \Rightarrow r$ | $(p \Rightarrow r \vee(q \Rightarrow r))$ | $S 2$ |
| $T$ | $T$ | $T$ | $T$ | $T$ | $T$ | $T$ | $T$ |
| $T$ | $T$ | $F$ | $F$ | $F$ | $F$ | $T$ | $T$ |
| $T$ | $F$ | $T$ | $T$ | $T$ | $T$ | $T$ | $T$ |
| $F$ | $T$ | $T$ | $T$ | $T$ | $T$ | $T$ | $T$ |
| $F$ | $T$ | $F$ | $F$ | $F$ | $F$ | $T$ | $F$ |
| $F$ | $T$ | $T$ | $T$ | $T$ | $T$ | $T$ | $T$ |
| $F$ | $F$ | $F$ | $T$ | $T$ | $T$ | $T$ | $T$ |
$S 2 \rightarrow$ not a tautology
Standard 11
Mathematics
