Question

Consider,
Statement – 1: $\left( {p\wedge \overrightarrow q} \right)\wedge \left( {\overrightarrow p\wedge q} \right)$ is a fallacy.
Statement – 2: $\left( {p \to q} \right) \leftrightarrow \left( {\overrightarrow q \to \overrightarrow p} \right)$ is a tautology.
A) Statement – 1 is false, Statement – 2 is true.
B) Statement – 1 is true, Statement – 2 is true and Statement – 2 is a correct explanation for Statement – 1.
C) Statement – 1 is true, Statement – 2 is true and Statement – 2 is not a correct explanation for Statement – 1.
D) Statement – 1 is true, Statement – 2 is false.

Answer 1

Hint: First find the truth table for the statement – 1 and check whether $\left( {p\wedge \overrightarrow q} \right)\wedge \left( {\overrightarrow p\wedge q} \right)$ is a fallacy or not. After that find the truth table for the statement – 2 and check whether $\left( {p \to q} \right) \leftrightarrow \left( {\overrightarrow q \to \overrightarrow p} \right)$ is a tautology or not. After that check whether the statement – 2 is an explanation of statement – 1 or not.

Complete step by step answer:
Check, the truth table for the statement – 1 is,

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

The truth table shows that statement – 1 is a fallacy. So, the statement – 1 is true.
Now check, the truth table for the statement – 2 is,

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

The truth table shows that statement – 2 is a tautology. So, statement – 2 is true.

Since, the statement – 1 and statement – 2 are different. So, statement – 2 is not an explanation of the statement – 1.

Therefore, option (C) is correct.

Note:
The students might make mistakes by not checking the statement with the truth table.
A truth table is a tabular representation of all the combinations of values for inputs and their corresponding outputs. It is a mathematical table that shows all possible outcomes that would occur from all possible scenarios that are considered factual, hence the name.