Associative Property
If a non - empty set S is closed under a binary operation *, that is a*b € S. Then a binary operation is associative if (a*b) *c = a*(b*c)

Such that C also belongs to S.

Example: The operation Ө on the set Z of integers is defined by; a Ө b = 2a + 3b - 1. Determine whether or not the operation is associative in Z.

Solution

Introduce another element C

(aӨb)Өc = (2a+3b-1) Ө C

= 2(2a+3b-1) + 3c - 1

= 4a + 6b - 2 + 3c - 1

= 4a + 6b + 3c - 3

Also, the RHS, a Ө (b Ө c) = a Ө (2b + 3c - 1)

= 2a + 3(2b + 3c - 1) - 1

= 2a + 6b + 9c - 3 -1

a Ө (b Ө c) = 2a + 6b + 9c -4

Since, (a Ө b) Ө c ≠ a Ө (b Ө c), the operation is not associative in Z.