4. Are the following sets closed under the following operations? Prove your answer.
If a set is not closed under the operation , what i its closure under the operation?

a. L = {w E {a , b} * : W ends in a} under the function odds, defined on String
as folIows: odds(s) = the string that 1S formed by concatenating together all of the odd numbered characters of s, (Start numbering the character at 1.)
For ex, odds( ababbbb) = aabb.

b. FIN (the set of finite languages) under the function oddsL d fined n languages
as folIows:
oddsL(L) = {w: 3XE L(w = odds (x)) }.

c. INF (the set of infinite languages) under the function odd L.

d. FIN under the funetion maxslring, defined in E xample 8.22.

e. INF under the funetion maxstring.