Monday, March 28, 2022

A binary operation ⊕ on a set of integers is defined as x ⊕ y = x2 + y2 . Which one of the following statements is TRUE about ⊕?

Q1. A binary operation ⊕ on a set of integers is defined as x ⊕ y = x2 + y2 . Which one of the following statements is TRUE about ⊕?
(A) Commutative but not associative (B) Both commutative and associative
(C) Associative but not commutative (D) Neither commutative nor associative

Ans: option (A)
Explanation:
A binary operation is said to be commutative if changing the order of the operands does not change the result. For example: 
Addition is commutative => (5 + 4 = 9  and  4 + 5 = 9). Similarly multiplication also.
Subtraction is not commutative => (5 - 4 = 1  and  4 - 5 = -1). Similarly division is also not commutative.

Associative Property:
Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value. Consider, for instance, the following equations:

For example: Addition is said to have associative property.
(5 + 3) + 1 = 5 + (3 + 1) = 9

So coming back to our problem:
Checking Commutative Property:
x ⊕ y = x2 + y2
y ⊕ x = y2 + x2
Since addition is commutative x ⊕ y = y ⊕ x
Consider x=2 and y=3
x ⊕ y = y ⊕ x = 13

Checking Associative Property:
(x ⊕ y) ⊕ z = (x ⊕ y)2 + z2
x ⊕ (y ⊕ z) = x2 + (y ⊕ z)2

Considering x=2, y=3 and z=1
(2 ⊕ 3) ⊕ 1 = (2 ⊕ 3)2 + 12 = 132 +1 = 170
2 ⊕ (3 ⊕ 1) = 22 + (3 ⊕ 1)2 = 4 + 102 = 104

Therefore ⊕ operation satisfies commutative property but not associative property.


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