Project ForgeDiscrete Mathematics
Set Theory & Abstract Algebra
GATE 2013
Medium
A binary operation ⊕ on a set of integers is defined as x ⊕ y = x² + y². Which one of the following statements is TRUE about ⊕?
- Commutative but not associative
- BBoth commutative and associative
- CAssociative but not commutative
- DNeither commutative nor associative
Explanation
Commutative: x⊕y = x²+y² = y²+x² = y⊕x, always true. Associative: (x⊕y)⊕z = (x²+y²)²+z², while x⊕(y⊕z) = x²+(y²+z²)² — testing x=1,y=1,z=0 gives (1+1)²+0=4 vs 1+(1+0)²=2, which differ. So ⊕ is commutative but not associative.
Source: GATE CS 2013 (2013)
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