what's the easiest way to determine whether the binary operation * is associative, if * is defined by $x*y=\frac{x+y}{1+xy} $

Question

anonymous · Answer

Define associativity

sasogeek · Answer

x*(y*z)=(x*y)*z ... if it's associative, then the right hand side should be equal to the left hand side

anonymous · Answer

It is associative i did it by verifying LHS=RHS
Although intuition pointed it right away

sasogeek · Answer

well is that the easiest way...? i was thinking maybe there's a faster method

mr.math · Answer

It clearly follows from the associativity of addition and multiplication.

sasogeek · Answer

what do u mean by that?

mr.math · Answer

I mean $(x*y)*z=\frac{(x+y)+z}{1+(xy)z}=\frac{x+(y+z)}{1+x(yz)}=x*(y*z).$

sasogeek · Answer

ohhh okay, well i was doing some weird workings but the answer was quite obvious right from the beginning :) thanks

mr.math · Answer

You're welcome.