Is this operation associative?

HOME

Math Central

Quandaries & Queries

Question from Francesca, a student:

Determine whether the binary operation * defined is commutative and whether * is associative
* defined on Z by a*b = a-b\
I understand how to figure out if it's commutative, but I thought for a binary operation to be associative, it had to have at least three elements, so I don't know how to tell if this associative or not.

We have two responses for you

Hi Francesca,

I expect the definition of * is

For any integers a and b, a*b = a - b.

To check if this operation is associative let a, b and c be any integers and see if you can verify that

a*(b*c) = (a*b)*c

If you think it i not true you can verify that by finding specific integers a, b and c and showing that a*(b*c) ≠ (a*b)*c

If you need further help write back,
Penny

Francesca,

An operation * is associative if (a*b)*c = a*(b*c) for all possible a, b and c.
Does that work for your operation?

Victoria

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.