Question

Which of the following statements is a tautology?
A. $\left( { \sim q \wedge p} \right) \wedge q$
B. $\left( { \sim q \wedge p} \right) \wedge \left( {p \wedge \sim p} \right)$
C. $\left( { \sim q \wedge p} \right) \vee \left( {p \vee \sim p} \right)$
D. $\left( {p \wedge q} \right) \wedge \left( {p \wedge q} \right)$

Answer 1

Hint: We will first create all the possible values and everything in a table. After that, we will find the possible values of all the inside values inside the parenthesis.

Complete step-by-step solution:
Let us first consider the possibilities. We have two statements here, p and q. Both can be true and false both. So, we can 2 into 2 = 4 possibilities.
So, let us create a table for this:

p	q
T	T
T	F
F	T
F	F

Here, we now have the required table.
Option A: Now the first option given to us is: $\left( { \sim q \wedge p} \right) \wedge q$
Now, here we have two statements p and q.
Now, we will find the possibilities of $ \sim q$.

p	q	$ \sim q$
T	T	F
T	F	T
F	T	F
F	F	T

Now, let us add $ \sim q \wedge p$ in the table:-

p	q	$ \sim q$	$ \sim q \wedge p$
T	T	F	F
T	F	T	T
F	T	F	F
F	F	T	F

Now, we will just get the truth table for the first option that is $\left( { \sim q \wedge p} \right) \wedge q$ using this data in above table:-

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

Option B: Now we have the option $\left( { \sim q \wedge p} \right) \wedge \left( {p \wedge \sim p} \right)$.
For this we will use the table:

p	q	$ \sim q$	$ \sim q \wedge p$
T	T	F	F
T	F	T	T
F	T	F	F
F	F	T	F

Now, we will just have to find the value of $p \wedge \sim p$ :

p	q	$ \sim p$	$p \wedge \sim p$
T	T	F	F
T	F	F	F
F	T	T	F
F	F	T	F

Now let us combine these two tables to get the following table:-

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

Option C: Now we have the option $\left( { \sim q \wedge p} \right) \vee \left( {p \vee \sim p} \right)$
For this we will use the table:

p	q	$ \sim q$	$ \sim q \wedge p$
T	T	F	F
T	F	T	T
F	T	F	F
F	F	T	F

Now, we will just have to find the value of $p \vee \sim p$ :

p	q	$ \sim p$	$p \vee \sim p$
T	T	F	T
T	F	F	T
F	T	T	T
F	F	T	T

Now let us combine these two tables to get the following table:-

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

Option D: Now we have the option $\left( {p \wedge q} \right) \wedge \left( {p \wedge q} \right)$.
For this we will need to use the following table:-

p	q	$p \wedge q$
T	T	T
T	F	F
F	T	F
F	F	F

So, for this we will then get:-

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

Now, tautology is a statement which is always true.

Hence, the answer is option C) $\left( { \sim q \wedge p} \right) \vee \left( {p \vee \sim p} \right)$.

Note: The students must know that Tautology is a statement which is always true.
Here, we can clearly see that since in option C we have $p \vee \sim p$ which is no matter what is always going to be true always. Hence, we have the option C as a tautology.
The word “tautology” comes from the work “the same”.