Let = {w, x, y] and 2 = {x, y, z} be alphabets. If A1 =

Question:

Let ∑ = {w, x, y] and ∑2 = {x, y, z} be alphabets. If A1 = {xtyj|i, j ∈ Z+, j > i > 1}, A2 = {wtyJi, j ∈ Z+, i > j > 1}, A3 = {wlxJylzJ\i, j ∈ Z+, j > i > 1}, and A4 = {zJ (wz)lwJ\i, j ∈ Z+, i > 1, j > 2}, determine whether each of the following statements is true or false۔
(a) Ai is a language over ∑1.
(b) A2 is a language over ∑2.
(c) A3 is a language over ∑1 ∪ ∑2.
(d) A] is a language over ∑1 ∩ ∑2.
(e) A4 is a language over ∑1 △ ∑2.
(f) A] U A2 is a language over ∑1.