# Polytime Mapping Reduction from Language A to Language A (identity)

How would I create a polytime mapping reduction to prove A ≤p A for any language A.

I was thinking to assume A is in P to start. For every 𝑥: 𝑥∈𝐴 iff 𝑓(𝑥)∈𝐴.

But I am not sure what to do from there, any advice? I don't get how to show that you can polytime reduce a function to itself, because if you assume A has a Turing machine and a polytime decider, a function that is a polytime mapper, then how do you show its equal to each other?

• I don't understand your confusion. What do you mean by "A has a Turing machine"? Nov 10, 2021 at 2:45

You need to construct a polytime function $$f$$ with the following property:
For every $$x$$: $$x \in A$$ iff $$f(x) \in A$$.