Skip to content
Related Articles

Related Articles

Improve Article
GATE | GATE CS 2013 | Question 1
  • Difficulty Level : Easy
  • Last Updated : 17 Oct, 2013

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


Answer: (A)

Explanation:

Associativity:

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.

Commutativity:

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.

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 x⊕ y = x2 + y2.
for Commutativity: x ⊕y= y ⊕x.

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

for Associativity: x ⊕ (y ⊕ z) =(x ⊕ y) ⊕ z

LHS=> x ⊕ (y⊕ z) = x ⊕ ( y^2+z^2)= x^2+(y^2+z^2)^2

RHS=> (x ⊕y) ⊕z= ( x^2+y^2) ⊕z=(x^2+y^2)^2+z^2

So, LHS ≠ RHS, hence not associative.

Reference:
http://faculty.atu.edu/mfinan/4033/absalg3.pdf

This solution is contributed by Nitika Bansal

Another Solution :
\oplus commutative as x\oplusy is always same as y\oplusx.

\oplus is not associative as (x\oplusy)\oplusz is (x^2 + y^2)^2 + z^2, but x\oplus(y\oplusz) is x^2 + (y^2 + z^2)^2.


Quiz of this Question

Attention reader! Don’t stop learning now. Learn all GATE CS concepts with Free Live Classes on our youtube channel.

My Personal Notes arrow_drop_up
Recommended Articles
Page :