I have the following language:
$$L=\{\langle M1\rangle,\langle M2\rangle: |L(M_1) ∩ L(M_2)|≤10 ~~\text{ or }~~ |L(M_1) ∩ L(M_2)|≥1000\}$$
I am told that this language is either r,re or co-re. it was obvious that it is not in r.
I used a reduction to prove it is not in re.
My question is: how would one go about proving it is in co-re without reductions, or how would you go about creating a Turing machine that accepts the complement or rejects L.
(Note: to me this looks to not be in either re or co-re, but in the question, I am asked to identify it to be in either r, re or co-re).