Mathematical Reasoning
Mathematical Reasoning
nta_abhyas_2025
Grade 11
Question:
If $p$ and $q$ are two logical statements, then $p \iff q$ (as $p \Leftrightarrow q$) is equivalent to
$p \iff (p \Rightarrow q)$
$p \iff (p \vee q)$
$p \iff (p \wedge q)$
$p \iff (p \Rightarrow q)$
Step-by-Step Solution
Key Concept: A statement is a tautology if it is always true regardless of the truth values of its components
$p \Rightarrow (p \vee q)$ can be rewritten as $p \Rightarrow (q \Rightarrow p) = \neg p \vee (\neg q \vee p) = t$ (true). Since this is a tautology, $p \Rightarrow (p \vee q)$ is a tautology. Now $p \Rightarrow (q \Rightarrow q)$ is false when $p$ is true and $q$ is false. Also, $p \Rightarrow (p \Rightarrow q)$ is false when $p$ is true and $q$ is false. Hence, (B) is correct.
Correct Answer: 2