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. (i) On Z + , define * by a * b = a − b (ii) On Z + , define * by a * b = ab (iii) On R , define * by a * b = ab 2 (iv) On Z + , define * by a * b = | a − b | (v) On Z + , define * by a * b = a

Answer 1

(i)

Consider the values of a=1 and b=2 for the provided equation.

1*2=1−2 =−1

For the given case, such that a,b∈ Z + , the function provides the value that is not defined in its range.

Thus, the function * is not a binary operation.

(ii)

Consider the values of a=1 and b=2 for the provided equation.

1*2=1( 2 ) =2

For the given case such that a,b∈ Z + , the function provides the value that is defined in its range. So it is deduced that the function * have a unique element ab∈ Z + . It means that the function * carries each pair ( a,b ) to a unique element a*b=ab in Z + .

Thus, the function * is a binary operation.

(iii)

Consider the values of a=1 and b=2 for the provided equation.

1*2=1 ( 2 ) 2 =4

For the given case such that a,b∈R , the function provides the value that is defined in its range. So it is deduced that the function * have a unique element a b 2 ∈R . It means that the function * carries each pair ( a,b ) to a unique element a*b=a b 2 in R .

Thus, the function * is a binary operation.

(iv)

Consider the values of a=1 and b=2 for the provided equation..

1*2=| 1−2 | =| −1 | =1

For the given case such that a,b∈ Z + , the function provides the value that is defined in its range. So it is deduced that the function * have a unique element | a−b |∈ Z + . It means that the function * carries each pair ( a,b ) to a unique element a*b=| a−b | in Z + .

Thus, the function * is a binary operation.

(v)

Consider the values of a=1 and b=2 for the provided equation.

1*2=1 =1

For the given case such that a,b∈ Z + , the function provides the value that is defined in its range. So it is deduced that the function * have a unique element a∈ Z + . It means that the function * carries each pair ( a,b ) to a unique element a*b=a in Z + .

Thus, the function * is a binary operation.