###### Question 10033 – GATE 1997

April 30, 2024###### Question 8833 – Minimum-Spanning-Tree

April 30, 2024# Question 8737 – GATE 2013

A binary operation ⊕ on a set of integers is defined as x ⊕ y = x^{2 }+ y^{2}. Which one of the following statements is **TRUE** about ⊕?

Correct Answer: A

Question 1 Explanation:

Cumulative property:

A binary relation on a set S is called cumulative if a*b = b*a ∀ x,y∈S.

Associative property:

A binary relation on set is called associative if (a*b)*c = a*(b*c) ∀ x,y∈S.

Given x⊕y = x

Replace x, y in (1)

y⊕x = y

(x⊕y)⊕z = (x

= (x

= x

x⊕(y ⊕ z) = x ⊕ (y

= x

= x

A binary relation on a set S is called cumulative if a*b = b*a ∀ x,y∈S.

Associative property:

A binary relation on set is called associative if (a*b)*c = a*(b*c) ∀ x,y∈S.

Given x⊕y = x

^{2}+ y^{2}——–(1)Replace x, y in (1)

y⊕x = y

^{2}+ x^{2}which is same as (1), so this is cumulative(x⊕y)⊕z = (x

^{2}+ y^{2}) ⊕ z= (x

^{2}+ y^{2}) + z^{2}= x

^{2}+ y^{2}+ z^{2}+ 2x^{2}y^{2}———-(2)x⊕(y ⊕ z) = x ⊕ (y

^{2}+ z^{2})= x

^{2}+ (y^{2}+ z^{2})^{2}= x

^{2}+ y^{2}+ z^{2}+ 2y^{2z2 ———– (3) (2) & (3) are not same so this is not associative. }Commutative but not associative

Both commutative and associative

Associative but not commutative

Neither commutative nor associative

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