Question

Assertion
STATEMENT 1: $\sim \left( p\leftrightarrow \sim q \right)$ is equivalent to $\left( p\vee \sim q \right)\wedge \left( \sim p\vee q \right)$
Reason
STATEMENT 2: $\sim \left( p\leftrightarrow q \right)$ is equivalent to $\left( p\wedge \sim q \right)\vee \left( \sim p\wedge q \right)$
A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation for Statement 1
B) Statement 1 is True, Statement 2 is True; Statement 2 is NOT a correct explanation for Statement 1
C) Statement 1 is True, Statement 2 is False
D) Statement 1 is False, Statement 2 is True

Answer 1

Hint: To know which of the given options is correct we will start by checking the two statements. Firstly we will use the truth table for statement 1 left hand side and right hand side and check whether the answer obtained is the same for both. Then we will do the same process for statement 2. If both statements are correct, the reason statement is the correct explanation of the assertion statement.

Complete step-by-step solution:
Firstly starting with Assertion which is given as:
 STATEMENT 1: $\sim \left( p\leftrightarrow \sim q \right)$ is equivalent to $\left( p\vee \sim q \right)\wedge \left( \sim p\vee q \right)$
We will form the table for both the values separately as:
$\sim \left( p\leftrightarrow \sim q \right)$ Is given in below truth table

$p$	$q$	$\sim q$	$p\leftrightarrow \sim q$	$\sim \left( p\leftrightarrow \sim q \right)$
T	T	F	F	T
T	F	T	T	F
F	T	F	T	F
F	F	T	F	T

$\left( p\vee \sim q \right)\wedge \left( \sim p\vee q \right)$ Is given in below truth table:

$p$	$q$	$\sim p$	$\sim q$	$\left( p\vee \sim q \right)$	$\sim p\vee q$	$\left( p\vee \sim q \right)\wedge \left( \sim p\vee q \right)$
T	T	F	F	T	T	T
T	F	F	T	T	F	F
F	T	T	F	F	T	F
F	F	T	T	T	T	T

As we can see that both table last columns give the same value therefore Statement 1 is true.
Next we will check the reason statement which is given as follows:
STATEMENT 2: $\sim \left( p\leftrightarrow q \right)$ is equivalent to $\left( p\wedge \sim q \right)\vee \left( \sim p\wedge q \right)$
We will form the table for both the values separately as:
$\sim \left( p\leftrightarrow q \right)$ Is given in below truth table

$p$	$q$	$\left( p\leftrightarrow q \right)$	$\sim \left( p\leftrightarrow q \right)$
T	T	T	F
T	F	F	T
F	T	F	T
F	F	T	F

$\left( p\wedge \sim q \right)\vee \left( \sim p\wedge q \right)$ Is given in below truth table:

$p$	$q$	$\sim p$	$\sim q$	$\left( p\wedge \sim q \right)$	$\left( \sim p\wedge q \right)$	$\left( p\wedge \sim q \right)\vee \left( \sim p\wedge q \right)$
T	T	F	F	F	F	F
T	F	F	T	T	F	T
F	T	T	F	F	T	T
F	F	T	T	F	F	F

As we can see that both table last columns give the same value therefore Statement 2 is true.
Now as both Statements are true that mean Statement 2 is the correct explanation for Statement 1.
Hence option (A) is correct.

Note: Logic operations are those operations which change the Boolean values. Boolean values are either true or false. In order to form a truth table we form a row which is equal to ${{2}^{n}}$ where $n$ is the number of letters in each statement. There are various types of logic operators which are Conjunction, Negation, Disjunction, Conditional and Bi-conditional. All have different rules by which we solve the truth table.