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# GATE | GATE CS 2013 | Question 1

• Difficulty Level : Easy
• Last Updated : 28 Jun, 2021

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

Explanation:

Associativity:

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

a âˆ— (b âˆ—c) = (a âˆ—b) âˆ—c for all a, b, c âˆˆS.

Commutativity:

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

a âˆ—b=b âˆ—a for all a, b, âˆˆS.

In this case, the order in which elements are combined does not matter.

Solution:

Here a binary operation on a set of integers is defined as xâŠ• y = x2 + y2.
for Commutativity: x âŠ•y= y âŠ•x.

LHS=> x âŠ•y= x^2+ y^2
RHS=> y âŠ•x= y^2+x^2
LHS = RHS. hence commutative.

for Associativity: x âŠ• (y âŠ• z) =(x âŠ• y) âŠ• z

LHS=> x âŠ• (yâŠ• z) = x âŠ• ( y^2+z^2)= x^2+(y^2+z^2)^2

RHS=> (x âŠ•y) âŠ•z= ( x^2+y^2) âŠ•z=(x^2+y^2)^2+z^2

So, LHS â‰  RHS, hence not associative.

This solution is contributed by Nitika Bansal

Another Solution :
commutative as xy is always same as yx.

is not associative as (xy)z is (x^2 + y^2)^2 + z^2, but x(yz) is x^2 + (y^2 + z^2)^2.

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