If $P$ and $Q$ are two statements, then which of the following compound statement is a tautology?

$((P \Rightarrow Q) \land ~ Q) \Rightarrow P$

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$((P \Rightarrow Q) \land ~ Q) \Rightarrow ~ P$

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$((P \Rightarrow Q) \land ~ Q)$

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$((P \Rightarrow Q) \land ~ Q) \Rightarrow Q$

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Solution

The correct option is B
$((P \Rightarrow Q) \land ~ Q) \Rightarrow ~ P$

Explanation for correct option:
Option(B): $((P \Rightarrow Q) \land ~ Q) \Rightarrow ~ P$
Tautology: If each entry in the final column of the truth table of a compound statement is True alone, then the compound statement is called as a Tautology.
Now we verify whether the given statement is a tautology or not with the help of the Truth table.
$P$
$Q$
$~ P$
$~ Q$
$P \Rightarrow Q$
$(P \Rightarrow Q) \land ~ Q$
$((P \Rightarrow Q) \land ~ Q) \Rightarrow ~ P$
T
T
F
F
T
F
T
T
F
F
T
F
F
T
F
T
T
F
T
F
T
F
F
T
T
T
T
T
In the above table, we can see all the entries are True.
So, the compound statement $((P \Rightarrow Q) \land ~ Q) \Rightarrow ~ P$ is a Tautology.
Hence, option (B) is correct.
Explanation for incorrect options:
Option (A): $((P \Rightarrow Q) \land ~ Q) \Rightarrow P$
We will check with the help of Truth table.
$P$
$Q$
$~ Q$
$P \Rightarrow Q$
$(P \Rightarrow Q) \land ~ Q$
$((P \Rightarrow Q) \land ~ Q) \Rightarrow P$
T
T
F
T
F
T
T
F
T
F
F
T
F
T
F
T
F
T
F
F
T
T
T
F
In the above table, all the entries in the last column are not true.
So, the compound statement $((P \Rightarrow Q) \land ~ Q) \Rightarrow P$ is not a tautology.
Hence, option (A) is incorrect.
Option (C): $((P \Rightarrow Q) \land ~ Q)$
We will check with the help of Truth table.
$P$
$Q$
$~ Q$
$P \Rightarrow Q$
$(P \Rightarrow Q) \land ~ Q$
T
T
F
T
F
T
F
T
F
F
F
T
F
T
F
F
F
T
T
T
In the above table, all the entries in the last column are not true.
So, the compound statement $((P \Rightarrow Q) \land ~ Q)$ is not a tautology.
Hence, option (C) is incorrect.
Option (D): $((P \Rightarrow Q) \land ~ Q) \Rightarrow Q$
We will check with the help of Truth table.
$P$
$Q$
$~ Q$
$P \Rightarrow Q$
$(P \Rightarrow Q) \land ~ Q$
$((P \Rightarrow Q) \land ~ Q) \Rightarrow Q$
T
T
F
T
F
T
T
F
T
F
F
T
F
T
F
T
F
T
F
F
T
T
T
F
In the above table, all the entries in the last column are not true.
Therefore, the compound statement $((P \Rightarrow Q) \land ~ Q) \Rightarrow Q$ is not a tautology.
Hence, option (D) is incorrect.
Hence, the correct answer is an option (B).

Suggest Corrections

$P$	$Q$	$~ P$	$~ Q$	$P \Rightarrow Q$	$(P \Rightarrow Q) \land ~ Q$	$((P \Rightarrow Q) \land ~ Q) \Rightarrow ~ P$
T	T	F	F	T	F	T
T	F	F	T	F	F	T
F	T	T	F	T	F	T
F	F	T	T	T	T	T