# Show that NP∩coNP =∅

I know that P is a subset of NP, but I'm not sure what this tells me about P as it relates to coNP? I feel like this is how I should go about proving it, but I'm not sure how. Otherwise, I could find a language that is in both NP and coNP, but I'm not sure how to prove any examples of that.

I know that I can prove L and L complement both exists in P and therefore NP, but I don't know how to relate this to coNP.

• I think you need to carefully look at the definition of $coNP$ again - your second paragraph suggests you can show that $P \subseteq NP \cap coNP$ (which would be a correct statement). Mar 17 '20 at 1:40
• @LukeMathieson, incorrect here, surely? (I'm not quite sure that that's what the second paragraph says, though.) Mar 17 '20 at 3:21
• Please check the title: did you intend P∩coNP =∅? Mar 17 '20 at 6:18
• PRIME is both in NP and co-NP. As is COMPOSITE. (Which is taken as an argument that they are both unlikely to be NO-complete) Mar 17 '20 at 7:55