Let X be a set and define the binary operation ∆ on the subsets of X by A∆B := (A∩Bc)∪(B∩Ac). Prove that the set P(X) of all subsets of X is a group under the operation ∆.

Let X be any non-empty set. Show that the set of all bijectionsof X is a group under the operation of function composition. Isthis group abelian?

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

Let G1 and G2 be groups. Prove that G1 × G2 is isomorphic to G2 × G1.