1. The problem statement, all variables and given/known data So we have this operation x*y=x+2y+4 and then our 2nd one is x*y=x+2y-xy I need to check if it is commutative,associative, and if it has a identity and an inverse. 3. The attempt at a solution y*x=y+2x+4 so it is not commutative x*(y*z)=x+2(y+2z+4)+4=x+2y+4z+12 (x*y)*z=(x*y)+2z+4=x+2y+4+2z+4 Not associative Now I will solve for the identity e*x=e+2x+4=x e=-x-4 x*e=x+2e+4=x e=-2 Since I have 2 different identity elements this means that one does not exist because e should be unique. Since there is no e there is no inverse. Now for the second one x*y=x+2y-xy y*x=y+2x-yx Does not commute (x*y)*z=x+2y-xy+2z-xz-2yz+xyz x*(y*z)=x+2y+4z-2yz-xy-2xz+xyz Not associative x*e=x+2e-xe=x e=0 e*x=e+2x-ex=x e(1-x)=-x The identity element does not seem to be unique so it does not exist. Just want to know if I am doing this right.