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. (i) `O n Z^+, d efin e *b ya*b = a - b`(ii) `O n Z^+,

Explanation

In `Z^(+), a ** b =a - b`
`2**3 = 2-3 = -1 in Z^(+)`
`therefore` Given operation is not binary.
(ii) In `Z^(+), a ** b = ab`
The product of every two positive intergers is a positive interger.
`therefore` Given operations is binary.
(iii) In R, `a ** b= ab^(2)`
The square of every real number is always a real number.
(iv) In `Z^(+) , " "a**b = a`
The modulus of the difference of every two positive integer.
`therefore` Given operation in binary.
(v) `In Z^(+), " " a**b=a`
For each `a,b in Z^(+)`
`" "a * b = a in Z^(+)`
`therefore` Given operation is binary.