GATE | GATE CS 2013 | Question 1
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
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.
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.
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.
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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