Let Σ and ∆ be any two alphabets. In this question, we consider maps h : Σ∗ → ∆∗ that satisfy the following properties: (I) h(e) = e, (II) h(uv) = h(u)h(v) for any pair of words u, v ∈ Σ∗. Such maps are uniquely determined by their values on letters in Σ, since for w = a1 . . . an we have h(w) = h(a1 . . . an) = h(a1) . . . h(an) by property (II). For any language L ⊆ Σ∗, we define h(L) := {h(w) | w ∈ L} ⊆ ∆∗ For parts (a)–(c) below, we consider the following special case: Let Σ = {0, 1} and ∆ = {a, b}, and suppose that h : {0, 1}∗ → {a, b}∗ satisfies the properties (I) and (II), with h(0) = aa and h(1) = b. (a) Compute h(00), h(110) and h(10110).
