Question

Consider:
Statement 1 : $\left( p\wedge \sim q \right)\wedge \left( \sim p\wedge q \right)$ is a fallacy.
Statement 2 : $\left( p\to q \right)\leftrightarrow \left( \sim q\to \sim p \right)$ is a tautology.
(1) Statement 1 is true, statement 2 is false.
(2) Statement 1 is false, statement 2 is true.
(3) Statement 1 is true, statement 2 is true: Statement 2 is a correct explanation for statement 1.
(4) Statement 1 is true, statement 2 is true: Statement 2 is not the correct explanation for statement 1.

Answer 1

Hint: Draw a truth table for the expressions in both the statements. For statement 1, check if all the values of the expression are false or not. If they are false, then the statement will be a fallacy, otherwise not. For statement 2, check if all the values of the expression are true or not. If all are true then the statement will be a tautology, otherwise not.

Complete step-by-step solution:
Here, we have been provided with two statements. We have to check whether statement 1 is a fallacy or not and whether statement 2 is a tautology or not. So, let us check them one by one using the truth table.
(i) Statement 1 : $\left( p\wedge \sim q \right)\wedge \left( \sim p\wedge q \right)$ is a fallacy.
Drawing the truth table, we get,

$p$	$q$	$\sim p$	$\sim q$	$p\wedge \sim q$	$\sim p\wedge q$	$\left( p\wedge \sim q \right)\wedge \left( \sim p\wedge q \right)$
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

Clearly, we can see that all the values of the expression are false. Hence statement 1 is a fallacy.
(ii) Statement 2 : $\left( p\to q \right)\leftrightarrow \left( \sim q\to \sim p \right)$ is a tautology.
Here, we are going to use the following basic truth table :

$A$	$B$	$A\to B$	$A\leftrightarrow B$
T	T	T	T
T	F	F	F
F	T	T	F
F	F	T	T

Now, drawing the truth table for statement 2, we get,

$p$	$q$	$\sim p$	$\sim q$	$p\to q$	$\sim q\to \sim p$	$\left( p\to q \right)\leftrightarrow \left( \sim q\to \sim p \right)$
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

Clearly, we can see that all the values of the expression are true. Hence, statement 2 is a tautology.
Now, statement 1 and statement 2 are different from each other, so they are independent. Therefore, statement 2 cannot be the correct explanation of statement 1.
Hence, option (4) is the correct answer.

Note: One may note that the above expressions can also be solved without using a truth table. But it will be beneficial for us to solve using the truth table as it will help us in understanding the concept of Boolean algebra more deeply. Further, if we forget the formula, then also by using the basic truth tables, we can solve the above question.