For a given alphabet E, let a ∈ ∑ - with a fixed. Define the functions pa, sa, r:∑ -> ∑* and the function d: E+ → E* as follows:
(i) The prefix (by a) function: pa(x) = ax, x ∈ ∑*.
(ii) The suffix (by a) function: sa(x) = xa, x ∈ ∑*.
(iii) The reversal function: r(λ) = λ for x ∈ ∑+, if x = x1x2 ∙∙∙∙∙∙ xn-1xn where xi- ∈ ∑ for all 1 < i < n, then r(x) = xnxn-1 ∙∙∙∙∙∙ x2x1 = xR (as defined in Example 6.16).
iv) The front deletion function: for x ∈ ∑+, if x = x1x2x3 ∙∙∙∙∙∙ xn, then d{x) = x2x3 ∙∙∙∙∙∙ xn.
(a) Which of these four functions is (or are) one-to-one?
(b) Determine which of these four functions is (or are) onto. If a function is not onto, determine its range.
(c) Are any of these four functions invertible? If so, determine their inverse functions.
(d) Suppose that ∑ = {a, e, i, o, u}. How many words x in ∑4 satisfy r(x) = x? How many in ∑5? How many in ∑n, where n ∈ N?
(e) For x ∈ ∑*, determine
(d o pa)(x) and (r o d o r o sa)(x).
f) If E = {a, e, i, o, u] and B = {ae, ai, ao, 00, eio, eiouu} ⊂ ∑*, find r-1(B), pa~l(B), sa-l(B), and |d-l(B)|.