# About NP-completeness and reduction

Probably this is a basic question but I'm not sure how to finish this proof. I have a problem $$X$$ and I want to prove that it is possible to reduce $$X$$ to another problem $$Y$$. I know that $$Y$$ is NP-complete and that $$Y$$ can be reduced to $$X$$. Can I just conclude that $$X$$ is also NP-complete and thus $$X$$ can be reduced to $$Y$$? Is it necessary to show that $$X$$ is in NP?

You cannot conclude that.

Knowing that $$Y$$ is NP-complete and reduces to $$X$$ is only enough to conclude that $$X$$ is NP-hard.

To show that $$X$$ can be reduced to $$Y$$ you will have to either prove that $$X\in NP$$, or show a reduction directly (don't do this, it is probably harder than just solving $$X$$ itself).