1. Let Σ be an alphabet. Prove or disprove that for every language L over alphabet Σ, if L2 = L, then L+ = L.

Suppose L is a non empty regular language, over Σ, such that every w E L satisfies |w] =k, for some ke N. Show that there cannot exist a DFA M = (E, Q, 8, q0, F), where |Q| = k.

4. Consider alphabet Σ = {0,1} and language Lo₁ = {we*: w=0"1" for some nonnegative n€Z}. Prove or disprove that for each language Lover Σ, if Ln Lo₁ = 0 and L is regular, then L is finite.

If A is any language, let ?− 1 2 be the set of all first halves of strings in A so that ?− 1 2 = (x | for some y, |x| = |y| and xy ϵ A}. Show that if A is regular, the so is ?− 1 2 .

A language L over an alphabet Σ is co-finite, if Σ∗ \L is empty or finite. LetCOFNFA = {〈N 〉|N is a NFA accepting a co-finite language}.Show that COFNFA is decidable.

Consider alphabet Σ = {0,1} and language Lo₁ = {weΣ* : w=0"1" for some nonnegative n€ Z}. Prove or disprove that for each language L over Σ, if L n Lo₁ = 0 and L is regular, then L is finite.

If A is any language, let A 1/3 – 1/3 be the set of all strings in A with their middle thirds removed so that A 1/3 – 1/3 = {x z| for some y, |x| = |y = |z| and xyz ∈ A}. Show that, if A is regular, then A 1/3 – 1/3 is not necessarily regular.

Let Σ={a,b}. For each of the following languages over Σ, prove that the language is regular (a)  La​={w∈Σ∗:n0​(w)is divisible by 4 andwends in aab}

The alphabet of the language is (a, b, c): Use pumping lemma to prove that the language (abc n>0) is not a regular language (please make sure to write pumping lemma for regular languages in your proof).

Let A and B be two languages over the same alphabet (the alphabet does not matter). Suppose that A \ B and B \ A are finite. Then, prove that A is regular if and only if B is regular.