Question

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

Answer 1

Answer:

Step-by-step 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.