Problem: For positive real numbers x and y define x∗y=x+yx⋅y; then
Answer Choices:
A. ∗ is commutative but not associative
B. ∗ is associative but not commutative
C. ∗ is neither commutative nor associative
D. ∗ is commutative and associative
E. none of these
Solution:
x∗y=x+yxy=y+xyx+y∗x
and
(x∗y)∗z=x+yxy∗z=x+yxy+zx+yxyz=xy+xz+yzxyz
Similarly x∗(y∗z)=xy+xz+yzxyz, so "*" is both commutative and associative.