GATE | GATE-CS-2015 (Set 2) | Question 65

Consider two decision problems Q1, Q2 such that Q1 reduces in polynomial time to 3-SAT and 3-SAT reduces in polynomial time to Q2. Then which one of the following is consistent with the above statement?
(A) Q1 is in NP, Q2 is NP hard
(B) Q2 is in NP, Q1 is NP hard
(C) Both Q1 and Q2 are in NP
(D) Both Q1 and Q2 are in NP hard


Answer: (A)

Explanation:

Q1 reduces in polynomial time to 3-SAT 
==> Q1 is in NP

3-SAT reduces in polynomial time to Q2 
==> Q2 is NP Hard.  If Q2 can be solved in P, then 3-SAT
    can be solved in P, but 3-SAT is NP-Complete, that makes 
    Q2 NP Hard


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