Question

The following statement $(p \to q) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$is
1. Equivalent to $ \sim p \to q$
2. Equivalent to $p \to \sim q$
3. a fallacy
4. a tautology

Answer 1

Hint: In this question, we have to make the table of $(p \to q) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$this statement. First step is to open each part of given statement in different columns. Now, take four conditions $(T,T),(T,F),(F,T),(F,F)$for $p$and $q$ column and using table for logical implication solve further.

Formula Used:
Table for negation –

$P$	$ \sim P$
T	F
F	T

Table for logical implication –

$P$	$Q$	$P \to Q$
T	T	T
T	F	F
F	T	T
F	F	T

Complete step by step Solution:
Table of the given statement $(p \to q) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$ is attached below,

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

As, all the values of statement $(p \to q) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$ are true.
Therefore, $(p \to q) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$ is a tautology.
$ \Rightarrow $Option (4) is the correct answer.

Hence, the correct option is 4

Note: In mathematics, a tautology is a compound statement that always yields the True value. The result in tautology is always true, regardless of what the constituent parts are. Contradiction or fallacy are the polar opposites of tautology.