NP-Completeness and commutative property If $X$ is NP-complete and for some $Y, X\leq_p Y$ and $Y\leq_p X$ what can we say about $Y$?

My intuition says that this is only the case when $X=Y$ but I'm not sure how to justify this.
Your property holds for, for example 3-SAT and 3-COLOURABILITY so it's certainly not required that $X=Y$.

The fact that $Y\leq_p X$ and $X\in\mathrm{NP}$ tells you that $Y\in\mathrm{NP}$; the fact that $X$&nbsp;is $\mathrm{NP}$-complete and $X\leq_p Y$ tells you that $Y$&nbsp;is $\mathrm{NP}$-hard.  Therefore, $Y$&nbsp;is $\mathrm{NP}$-complete.