GATE | GATE-CS-2006 | Question 27

Consider the following propositional statements:
P1 : ((A ∧ B) → C)) ≡ ((A → C) ∧ (B → C))
P2 : ((A ∨ B) → C)) ≡ ((A → C) ∨ (B → C))
Which one of the following is true?
(A) P1 is a tautology, but not P2
(B) P2 is a tautology, but not P1
(C) P1 and P2 are both tautologies
(D) Both P1 and P2 are not tautologies


Answer: (D)

Explanation: The easiest way to solve this question by creating truth tables for the expressions given. Note that P1 will be a tautology if truth table for left expression is exactly same as truth table for right expression. Same holds for P2 also.

 

A	B	C	((A ∧ B) → C))	((A → C) ∧ (B → C))	((A ∨ B) → C))	((A → C) ∨ (B → C))
0	0	0	T	T	T	T
0	0	1	T	T	T	T
0	1	0	T	F	F	T
0	1	1	T	T	T	T
1	0	0	T	F	F	T
1	0	1	T	T	T	T
1	1	0	F	F	F	F
1	1	1	T	T	T	T

So as we see from table, none of the P1 or P2 are tautologies, so option (D) is correct.

Source: www.cse.iitd.ac.in/~mittal/gate/gate_math_2006.html

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