Question

The statements $p \to \left( {q \to p} \right)$ is equivalent to
(a) $p \to \left( {p \vee q} \right)$
(b) $p \to \left( {p \wedge q} \right)$
(c) $p \to \left( {p \leftrightarrow q} \right)$
(d) $p \to \left( {p \to q} \right)$

Answer 1

We will construct truth tables for the given expression and also the truth tables for the given option. The option having the same truth table with the truth table of the given expression is equivalent to the given expression.

Complete step-by-step answer:
We have to find an equivalent expression for the expression $p \to \left( {q \to p} \right)$
We know that the two expressions are equivalent if they have the same truth table.
Hence, we will first draw the truth table of $p \to \left( {q \to p} \right)$
The conditional statement $p \to q$ is true in every case except when $p$ is a true statement and $q$ is a false statement.

$p$	$q$	$\left( {q \to p} \right)$	$p \to \left( {q \to p} \right)$
T	T	T	T
T	F	T	T
F	T	F	T
F	F	T	T

Similarly, we will construct a truth table for each of the given options.
$p \to \left( {p \vee q} \right)$
We know that $p \vee q$ is false only when both $p$ and $q$ are false.

$p$	$q$	$\left( {p \vee q} \right)$	$p \to \left( {p \vee q} \right)$
T	T	T	T
T	F	T	T
F	T	T	T
F	F	F	T

Here, we can see the truth tables match with the truth table of the given expression.
Therefore, $p \to \left( {q \to p} \right)$ is equivalent to $p \to \left( {p \vee q} \right)$.
Thus, option A is correct.

Note: The conditional statement $p \to q$ is true in every case except when $p$ is a true statement and $q$ is a false statement. $p \vee q$ is false only when both $p$ and $q$ are false. $p \wedge q$ is true only when both $p$ and $q$ are true. $p \leftrightarrow q$ is true only when $p$ and $q$have the same value