Question

A binary operation ⊕ on a set of integers is defined as x⊕ y = x² + y². Which one of the following statements is TRUE about ⊕?
1. Commutative but not associative
2. Both commutative and associative
3. Associative but not commutative 
4. Neither commutative nor associative

Answer 1

Correct Answer - Option 1 : Commutative but not associative

Concept:

Associative

A binary operation ∗ on a set S is said to be associative if it satisfies the associative law:

a ∗ (b ∗c) = (a ∗ b) ∗ c for all a, b, c ∈S.

Commutative

A binary operation ∗ on a set S is said to be commutative if it satisfies the condition:

a ∗b=b ∗a for all a, b, ∈S.

Calculations:

First started with checking for associative, x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z.

L.H.S: x ⊕ (y ⊕ z) = x ⊕ (y² + z²) = x² + (x² + z²)² .

R.H.S: (x ⊕ y) ⊕ z = (x² + y²) ⊕ z = (x² + y²)² + z² .

L.H.S ≠ R.H.S then the operator is not associative.

For commutative, x ⊕ y = y ⊕ x

L.H.S:  x ⊕ y = x² + y²

R.H.S: y ⊕ x = y² + x²  

L.H.S = R.H.S then the operator is Commutative.