Question

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

On Z⁺, define * by a * b = |a − b|

Answer 1

On Z⁺, * is defined by a * b = |a − b|.

It is seen that for each a, b ∈ Z⁺, there is a unique element |a − b| in Z⁺.

This means that * carries each pair (a, b) to a unique element a * b = 
|a − b| in Z⁺.

Therefore, * is a binary operation.