Question

Which one of the following Boolean expressions is a tautology?
1. (p ∧ q) ∨ (p ∧ ~q)
2. (p ∨ q) ∨ (p ∨ ~q)
3. (p ∨ q) ∧ (p ∨ ~q)
4. (p ∨ q) ∧ (~p ∨ ~q)

Answer 1

Correct Answer - Option 2 : (p ∨ q) ∨ (p ∨ ~q)

A tautology is a proposition that is always true.

p	q	~p	~q	p ∧ q	p ∨ q	p ∨ ~q	P ∧ ~q	~p ∨ ~q
T	T	F	F	T	T	T	F	F
T	F	F	T	F	T	T	T	T
F	T	T	F	F	T	F	F	T
F	F	T	T	F	F	T	F	T

 

Option (a):

p ∧ q	p ∧ ~q	(p ∧ q) ∨ (p ∧ ~q)
T	F	T
F	T	T
F	F	F
F	F	F

 

Option (b):

(p ∨ q)	(p ∨ ~q)	(p ∨ q) ∨ (p ∨ ~q)
T	T	T
T	T	T
T	F	T
F	T	T

 

Option (c):

(p ∨ q)	(p ∨ ~q)	(p ∨ q) ∧ (p ∨ ~q)
T	T	T
T	T	T
T	F	F
F	T	F

 

Option (d):

(p ∨ q)	(~p ∨ ~q)	(p ∨ q) ∧ (~p ∨ ~q)
T	F	F
T	T	T
T	T	T
F	T	F

 

From the given options, the table in option (b) is always true.

Thus, the option (b) is tautology.