Question

Is the set \mathbb{Z} a group under the operation a*b = a - b + ab ? Justify your answer.

Answer 1

Explanation:

Given that *is the binary operation in the sets of integers.

`a*b = a - b + ab`

closure: a-b+ab is again an integer belongs to Z. Hence closure is true.

Associativity:
`a*(b*c) = a*(b+c-bc)\\= a-b-c+bc+ab+ac-abc\\= a-b-c +ab+bc+ca-abc`

`(a*b)*c=(a+b-ab)*c\\=a+b-ab-c+ac+bc-abc\\`

The two are not equal. Hence this cannot be a group as associtiavity does not hold good.