In this problem, we’ll use the following definitions of operations on languages (sets of strings) a, b:

union a ∪ b = {x | x ∈ a or x ∈ b}
concatenation a ◦ b = {xy | x ∈ a and y ∈ b}
star a* = {x1x2 . . xk | k ∈ z and k ≥ 0 and each xi ∈ a}
for each of the following sets of strings over the alphabet {a, b}, answer the following questions:
(1) is ε (the empty string) in the set?
(2) what is an example of a string over {a, b} of length at least 2 that is in the set (or why isn’t there such an example)?
(3) what is an example of a string over this alphabet of length at least 2 that is not in the set (or why isn’t there such an example)?
(a) { w | w ∈ {a, b}*}
(b) { w | w ∈ {ab}*}
(c) { a, aa } ◦ { b, bb }
(d) { a } ∪ { aa } ∪ { ab } ∪ { ba }