I have to prove or disprove the following statements.

**a) ≈(L(X)) ⊆ ≈(L(Y)) ⇒ ~(X) ⊆ ~(Y)**

**b) L(X) ⊆ L(Y) ⇒ ~(X) ⊆ ~(Y)**

`Definitions: DFA:M=(K,Σ,q0,δ,F) x~y : ∃q∈Q (q0,x) |-- (q,λ) and (q0,y) |-- (q,λ) x≈y : ∀z∈Σ* xz∈L ⇔ yz∈L `

My problem is the formal proof.

I could show that L(X) ⊆ L(Y) ⇒ ~(X) ⊆ ~(Y) with the help of an exmaple, but that doesn’t show that it applies to all.

And the main problem is to show ~(X) ⊆ ~(Y) or ≈(L(X)) ⊆ ≈(L(Y)), because I don’t really get what that means and when or why it is a subset.

Maybe on of you has an idea to prove or disprove the statements.