On the set of non-negative integers, the binary operation ______ is associative and non-commutative. (Fill in a binary operation that satisfies both properties.)

GATE 1994 · Discrete Mathematics · Binary Operation · medium

Answer: a * b = a (left projection) is associative and non-commutative; equivalently a * b = b (right projection) also works.

1. Check associativity of a * b = a: (a*b)*c = a*c = a and a*(b*c) = a*b = a; both groupings equal a, so the law holds for all a,b,c
2. Disprove commutativity: a*b = a but b*a = b; taking a = 2, b = 5 gives 2*5 = 2 while 5*2 = 5, which are different