# Prove NP-completeness for union of NP-complete language and language in P

Given disjoint languages $X$ and $Y$ , where $X$ is NP-complete and $Y\in P$ , how do I prove that $X\cup Y$ is NP-complete?

My idea is to prove that $(X\cup Y)\in NP$ and then prove that $X\cup Y$ is NP-hard:

I can prove $(X\cup Y)\in NP$ by saying that since both are in NP, they each have polytime verifiers. Thus we can have a polytime verifier that on any input will use these verifiers and accept if any accept, reject otherwise.

I am stuck on the part of proving NP-hardness. The idea of arbitrary languages is throwing me off; I am trying to reduce a NP-complete problem (SAT for example) to $X\cup Y$ and using the fact that $X$ has some method to reduce to it already but I am lost as for what to do with $Y$ . I am thinking that given an input for SAT, I need to somehow change the input so that I can relate acceptance in $X\cup Y$ to acceptance in SAT; same for rejection.

Any guidance would be appreciated; thanks!