Consider an alphabet Σ. This question concerns the equivalence relation ≈L, for L ⊆ Σ∗ .
(a) Give a language L1 such that, for any u, v ∈ Σ ∗ , we have u ≈L1 v.
(b) Give a language L2 such that, for any u, v ∈ Σ ∗ with u != v, |u| = |v|, we have u !≈L2 v
Sorry I couldn't copy the non-equal or non-equivalence side over from the text so I added the !.
I believe to answer this question I'm supposed to make a language then think of two strings, u and v, that both satisfy the language. For (a) I tried doing Σ = {a, b}, L1 = {w ∈ Σ∗: |w| is even}, u = [ε, aa, ab, ba, aaaa, ...], v = [ε, aa, bb, aabb, ...]. Now I'm not too sure what to do for (b) because u cannot equal v. Any tips would be helpful.