Question

Equivalence proposition of $p \Leftrightarrow q$ is

Answer 1

Hint: If there are two statements such as $p$ and $q$ then the compound statement $\left( {p \Rightarrow q} \right) \wedge \left( {q \Rightarrow p} \right)$ means that $p$ implies $q$ and $q$ implies $p$, this is called a Bi-conditional statement or Equivalence. It is denoted by $p \Leftrightarrow q$ or $p \equiv q$. For two propositions to be Logically Equivalent they should have the identical truth tables.
Symbols used and their meanings:
$p \Rightarrow q$ means p implies q and it is a Conditional connective.
$p \wedge q$ means p and q and it is a Conjunction connective.
$p \vee q$ means p or q and it is a Disjunction connective.
$p \Leftrightarrow q$ means p if and only if q and it is a Biconditional connective.


Complete step by step solution
Given:
$p$ and $q$ are two statements then using the Logical Equivalences Involving Bi-conditional Statements $p \Leftrightarrow q$ can be written as:
$\left( {p \Rightarrow q} \right) \wedge \left( {q \Rightarrow p} \right)$
Now we have to prepare a truth table for $p \Leftrightarrow q$ on the basis of truth tables for $ \Rightarrow $ and $ \wedge $ which includes all the variables.

$p$	$q$	$p \Leftrightarrow q$	$\left( {p \vee q} \right) \vee \left( {p \wedge q} \right)$	$\left( {p \Rightarrow q} \right) \wedge \left( {p \Rightarrow q} \right)$	$\left( {p \wedge q} \right) \vee \left( {p \Rightarrow q} \right)$	\[\left( {p \wedge q} \right) \Rightarrow \left( {p \vee q} \right)\]
T	T	T	T	T	T	T
T	F	F	T	F	F	T
F	T	F	T	T	T	T
F	F	T	F	T	T	T

From the truth table it is clear that the propositions are logically equivalent. So, the logically equivalent proposition of $p \Leftrightarrow q$ is $\left( {p \wedge q} \right) \Rightarrow \left( {p \vee q} \right)$.

Note: It may be noted that from the table that $p \Leftrightarrow q$ is true only when either both $p$ and $q$ are true or both are false. Also, from the truth tables it is confirmed that the given propositions are logically equivalent because they have the identical truth tables.