Question

Determine which of the following binary operation is commutative and which is associative :

(i) * on N defined as a*b = 2^ab

(ii) * on N defined as a*b = a + b + a^ab

(iii) * on Z defined as a*b = a – b

(iv) * on Q defined as a*b = ab + 1

(v) * on R defined as a*b = a + b – 7

Answer 1

(i) Given a*b = 2^ab

Commutativity: Let a, b ∈ N

a*b = 2^ab

= 2^b.a

= b*a

So, a*b = b*a

∴ * is a commutative operation.

Associativity: Let a, b, c ∈ N

(a*b)*c = 2(ab)*2^c = 2^{ab + c}

= 2^c*2^(ab) = 2^{c +ab}

a*(b*c) = 2a*2^(bc) = 2^{a + bc}

2^ab+c ≠ 2^a+bc

It is clear that (a*b)*c ≠ a*(b*c)

So, (a*b)*c is not an associative operation.

Hence, a*b = 2^ab is commutative but not

associative.

(ii) Given a*b= a + b + a²b

Commutativity: Let a, b ∈ N

a*b = a + b + a²b

b*a = b + a + b²a

a*b ≠ b*a .

So, * is not a commutative operation.

Associativity: Let a, b, c ∈ N

(a*b)*c = (a + b + a²b)*c

a*(b*c) = a*(b + c + b²c)

It is clear that (a*b)*c ≠ a*(b*c)

So * is not an associative operation.

Hence, a*b = a + b + a²b is neither commutative nor associative.

(iii) Given, a*b = a – b

Commutativity:

a*b = a – b, (a, b ∈ Z)

b*a = b – a, (a, b ∈ Z)

a*b* ≠ b*a

So * is not a commutative operation.

Associativity :

(a*b)*c = (a – b)*c

= a – b-c

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

= a – b + c

∵ (a*b)*c ≠ a*(b*c)

So, it is not associative operation.

It is clear that

a*b = a – b is neither commutative nor associative.

(iv) Given, a*b = ab + 1

Commutativity: Let a, b ∈ Q

a*b = ab + 1 and : b*a= ba + 1

⇒ a*b= b*a

∴ It is commutative.

∴ Addition and multiplication of rational number is commutative.

Associativity: Let a, b, c ∈ Q

(a*b)*c = (ab + 1)*c

= ab + 1 + c

(b*c)*a = (bc + 1) +a

= (a*b)*c ≠ (b*c)*a

So, * is not associative.

It is clear from above that a*b = ab + 1 is commutative but not associative.

(v) Given, a*b = a + b – 7

Commutativity: In R,

a*b = a + b – 7

= b + a – 7

= b*a’

Associativity :

(a*b)*c = (a + b – 7)*c

= (a + b – 7) + c – 7

= a + b + c – 14

a*(b*c) = a*(b + c – 7)

= a + (b + c – 7) – 7

= a + b + c – 14

So, (a*b)*c = a*(b*c)

Hence, it is clear that a*b = a + b – 7 are

commutative and associative.