Ans is A and D.

0 votes

7. Which of the following statements are known to be true.

A. P is a subset of NP.

B. NP is a subset of P.

C. P is not a subset of NP.

D. NP is not a subset of P.

A. P is a subset of NP.

B. NP is a subset of P.

C. P is not a subset of NP.

D. NP is not a subset of P.

2 votes

Best answer

**Answer is ONLY A.**

There maybe Two scenarios :

1. If $P = NP$ then A and B will be True.

2. If $P \neq NP$ then A and D will be True.

Thus, Since it is NOT known whether $P = NP$ or Not. So, We can derive one thing for sure that P is a subset of NP for sure whatever be the case ($P = NP$ or Not)

Saying P is a Proper subset of NP would again be wrong.

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Deepak , I have one doubt.. P=NP implies P ⊆ NP and NP ⊆ P...and *P* ≠ *N**P *implies P ⊂ NP...So ,I think from these 2 things , we can only say P ⊂ NP..ie. P is a proper subset of NP , not P is subset of NP...Please correct me where I m wrong

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Take Contradiction Approach to see this. Let's say P is a Proper Subset of NP. Now take the case of "If $P = NP$, Now saying P is a Proper subset of NP would be wrong. But Still, Saying P is a Subset of NP is still correct Because A set is always a Subset of itself (But a set is never a Proper Subset of Itself).

If $P \neq NP$ then We can say both statements " Proper Subset" and "Subset". So, Saying "Just Subset" is True in both cases But Saying Proper Subset would be wrong, has been $P = NP$.

If $P \neq NP$ then We can say both statements " Proper Subset" and "Subset". So, Saying "Just Subset" is True in both cases But Saying Proper Subset would be wrong, has been $P = NP$.

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Deepak , I have one doubt.. P=NP implies P ⊆ NP and NP ⊆ P...and *P* ≠ *N**P *implies P ⊂ NP...So ,I think from these 2 things , we can only say P ⊂ NP..ie. P is a proper subset of NP , not P is subset of NP...

The reason why You got confused is the definition of "Set Equality" which You used i.e. P ⊆ NP and NP ⊆ P

(It is indeed a correct definition) And From this definition and P ⊂ NP, You took what is Common and the common thing that you found was "⊂". But it is wrong. Because What You should have considered was "P ⊆ NP and NP ⊆ P" But You only considered "P ⊆ NP".

There is Nothing common between "Set Equality" and "Proper Subset". Both are Mutually Exclusive and Exhaustive. But The Term "Subset" can be used for Both.