smc

First, let's check for commutivity. We have $$y\ast x=\frac{y\cdot x}{y+x}=\frac{x\cdot y}{x+y}=x\ast y$$, so $\ast$ is commutative. Now we check for associativity. We have $$(x\ast y)\ast z=\left(\frac{x\cdot y}{x+y}\right)\ast z=\frac{\frac{x\cdot y}{x+y}\cdot z}{\frac{x\cdot y}{x+y}+z}=\frac{x\cdot y\cdot z}{x\cdot y+z(x+y)}=\frac{x\cdot y\cdot z}{x\cdot y+y\cdot z+x\cdot z}.$$ Also, $$x\ast (y\ast z)=x\ast \left(\frac{y\cdot z}{y+z}\right)=\frac{x\cdot \frac{y\cdot z}{y+z}}{x+\frac{y\cdot z}{y+z}}=\frac{x\cdot y\cdot z}{x(y+z)+y\cdot z}=\frac{x\cdot y\cdot z}{x\cdot y+y\cdot z+x\cdot z}=(x\ast y)\ast z,$$ and so $\ast$ is also associative. $\boxed{\text{“*" is commutative and associative}}$