Explanation:

To understand the answer, let's first define what it means for a problem to be NP-complete, NP-hard, and in NP.

- NP-complete: A problem is NP-complete if it is both in NP (nondeterministic polynomial time) and every problem in NP can be reduced to it in polynomial time. In other words, if we can solve an NP-complete problem in polynomial time, then we can solve every problem in NP in polynomial time.

- NP-hard: A problem is NP-hard if every problem in NP can be reduced to it in polynomial time. NP-hard problems do not necessarily need to be in NP themselves.

- NP: A problem is in NP if a solution to it can be verified in polynomial time. NP stands for nondeterministic polynomial time, which means that there may exist an algorithm that can solve the problem in polynomial time, but we cannot guarantee it.

Explanation of Options:

a) If X can be solved in polynomial time, then so can Y: This statement is not necessarily true. Just because X reduces to Y in polynomial time does not mean that Y can be solved in polynomial time. It only means that if we can solve Y in polynomial time, then we can also solve X in polynomial time.

b) X is NP-complete: This statement is true. If X reduces to Y in polynomial time and Y is NP-complete, then X must also be NP-complete. This is because if we can solve Y in polynomial time, we can solve every problem in NP in polynomial time, including X.

c) X is NP-hard: This statement is not necessarily true. X is only NP-hard if every problem in NP can be reduced to it in polynomial time. We know that X reduces to Y in polynomial time, but this does not imply that every problem in NP can be reduced to X.

d) X is in NP, but not necessarily NP-complete: This statement is also not necessarily true. We know that X reduces to Y in polynomial time, which means that if we have a solution to Y, we can verify it for X in polynomial time. However, this does not necessarily mean that X is in NP. It may be possible that X is not in NP.

Therefore, the correct answer is option 'b'. X is NP-complete.

X is not np card how can it be np complete ??