Please explain your work a By applying all axioms of Group s

Please explain your work.

a) By applying all axioms of Group, show that the set of all integers Z = {.., -3, -2, -1, 0, 1, 2, 3, …} and arithmetic addition ‘+’ forms a Group.

b) Let ‘x’ denote the arithmetic multiplication operation on elements of set Z defined in a above.

Solution

Group means a non empty set with a binary operation, it need to satisfies the following axioms,

a) Let Z={…,-3,-2,-1,0,1,2,3,…} be a non empty set of integers. (Z,+) is an algebraic structure.

Since, for any a,b Z,

a+b Z

Hence (Z,+) satisfies Closure property.

            2) For any a,b,c Z,

a+(b+c)= (a+b)+c

Hence (Z,+) satisfies Associative law.

            3)   For any a Z, 0(is an identity) Z ,

Since a+0 = 0+a = a.

Hence 0, is an identity element in (Z,+)

               4)For any a Z, -a Z, such that a+(-a) = (-a)+a= 0 (identity)
                  Hence –a is an identity element of a, in (Z,+).

Therefore(Z,+) is a group.

b) (Z, X) is algebraic structure, clearly X is a binary operation on Z.

(Z,X) satisfies Closure, Associative, and existence of inverse properties.

Where multiplicative identity of (Z,X) is 1.

But for any a Z(0,1,-1) multiplicative inverse is not exist.

Hence (Z,X) is not a group.