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
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 BansalAnother 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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