Question

For each binary operation * defined below, determine whether * is commutative or associative. (i) On Z , define a * b = a − b (ii) On Q , define a * b = ab + 1 (iii) On Q , define a * b (iv) On Z + , define a * b = 2 ab (v) On Z + , define a * b = a b (vi) On R − {−1}, define

Answer 1

(i)

On the integer range that is Z , * function is defined by a*b=a−b .

Consider the values of a=1 , b=2 and c=3 .

The value of a*b .

a*b=1*2 =1−2 =−1

The value of b*a .

b*a=2*1 =2−1 =1

For the provided case such that a,b∈Z , the function provides different values for the operations a*b and b*a .

Thus, the function * is not commutative.

The value of the equation ( a*b )*c .

( a*b )*c=( 1*2 )*3 =( 1−2 )*3 =−1−3 =−4

The value of the equation a*( b*c ) .

a*( b*c )=1*( 2*3 ) =1*−1 =1−( −1 ) =2

For the provided case such that a,b,c∈Z , the function provides different values for the operations ( a*b )*c and a*( b*c ) .

Thus, the function * is not associative.

(ii)

On the positive integer range that is Q , * function is defined by a*b=ab+1 .

Consider the value of a=1 , b=2 and c=3 .

The value of a*b .

a*b=1*2 =1( 2 )+1 =3

The value of b*a .

b*a=2*1 =2( 1 )+1 =3

For the provided case such that a,b∈Q , the function provides same values for the operations a*b and b*a .

Thus, the function * is commutative.

The value of the equation ( a*b )*c .

( a*b )*c=( 1*2 )*3 =( 1( 2 )+1 )*3 =3( 3 )+1 =10

The value of the equation a*( b*c ) .

a*( b*c )=1*( 2*3 ) =1*( 7 ) =1( 7 )+1 =8

For the provided case such that a,b,c∈Q , the function provides different values for the operations ( a*b )*c and a*( b*c ) .

Thus, the function * is not associative.

(iii)

On the positive integer range that is Q , * function is defined by a*b= ab 2 .

Consider the value of a=1 , b=2 and c=3 .

The value of a*b .

a*b=1*2 = 1( 2 ) 2 =1

The value of b*a .

b*a=2*1 = 2( 1 ) 2 =1

For the provided case such that a,b∈Q , the function provides same values for the operations a*b and b*a .

Thus, the function * is commutative.

The value of the equation ( a*b )*c .

( a*b )*c=( 1*2 )*3 =1*3 = 3( 1 ) 2 = 3 2

The value of the equation a*( b*c ) .

a*( b*c )=1*( 2*3 ) =1*( 3 ) = 1( 3 ) 2 = 3 2

For the provided case such that a,b,c∈Q , the function provides same values for the operations ( a*b )*c and a*( b*c ) .

Thus, the function * is associative.

(iv)

On the positive integer range that is Z + , * function is defined by a*b= 2 ab .

Consider the value of a=1 , b=2 and c=3 .

The value of a*b .

a*b=1*2 = 2 ( 1 )( 2 ) = 2 2 =4

The value of b*a .

b*a=2*1 = 2 ( 2 )( 1 ) = 2 2 =4

For the provided case such that a,b∈ Z + , the function provides same values for the operations a*b and b*a .

Thus, the function * is commutative.

The value of the equation ( a*b )*c .

( a*b )*c=( 1*2 )*3 = 2 2 *3 =4*3 = 2 ( 4 )( 3 ) = 2 12

The value of the equation a*( b*c ) .

a*( b*c )=1*( 2*3 ) =1*( 2 6 ) =1*64 = 2 ( 64 )( 1 ) = 2 64

For the provided case such that a,b,c∈ Z + , the function provides different values for the operations ( a*b )*c and a*( b*c ) .

Thus, the function * is not associative.

(v)

On the positive integer range that is Z + , * function is defined by a*b= a b .

Consider the value of a=1 , b=2 and c=3 .

The value of a*b .

a*b=1*2 = 1 2 =1

The value of b*a .

b*a=2*1 = 2 1 =2

For the provided case such that a,b∈ Z + , the function provides different values for the operations a*b and b*a .

Thus, the function * is not commutative.

Consider different values for a=2 , b=3 and c=4 .

The value of the equation ( a*b )*c .

( a*b )*c=( 2*3 )*4 = 2 3 *4 = 8 4 = 2 12

The value of the equation a*( b*c ) .

a*( b*c )=2*( 3*4 ) =2*( 3 4 ) = 2 81

For the provided case such that a,b,c∈ Z + , the function provides different values for the operations ( a*b )*c and a*( b*c ) .

Thus, the function * is not associative.

(vi)

On the integer range that is R−{ −1 } , * function is defined by a*b= a b+1 .

Consider the value of a=1 , b=2 and c=3 .

The value of a*b .

a*b=1*2 = 1 2+1 = 1 3

The value of b*a .

b*a=2*1 = 2 1+1 =1

For the provided case such that a,b∈R−{ −1 } , the function provides different values for the operations a*b and b*a .

Thus, the function * is not commutative.

The value of the equation ( a*b )*c .

( a*b )*c=( 1*2 )*3 = 1 3 *3 = 1 3 3+1 = 1 12

The value of the equation a*( b*c ) .

a*( b*c )=1*( 2*3 ) =1* 2 4 = 1 1 2 +1 = 2 3

For the provided case such that a,b,c∈R−{ −1 } , the function provides different values for the operations ( a*b )*c and a*( b*c ) .

Thus, the function * is not associative.