A binary operation (+) on a set of integers is defined as x (+) y = x^2 + y^2. 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
GATE 2013 · Discrete Mathematics · Binary Operation · easy
Answer: A. Commutative but not associative
- Test commutativity: x (+) y = x^2 + y^2 and y (+) x = y^2 + x^2. Since addition of numbers is symmetric, these are equal for all x, y, so (+) is commutative.
- Test associativity with a counterexample: Take x=1, y=2, z=3. Left: (1 (+) 2) (+) 3 = (1+4) (+) 3 = 5 (+) 3 = 25 + 9 = 34. Right: 1 (+) (2 (+) 3) = 1 (+) (4+9) = 1 (+) 13 = 1 + 169 = 170.
- Combine the findings: Commutative (Step 1) but not associative (Step 2) selects option A.