Prove that in a group, (a-1)-1 = a for all a.

Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system (Zodd, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative. Is (Zodd, +) a semi-group?  Explain Is (Zodd, +) a group?  Explain

Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

For any two elements x and y in a group G, there is a unique elementz in G such that y = xz.