Given the law of composition xyxy2ax2y determine a and b if

Given the law of composition x*y=xy+2ax+2y determine a and b if the law is communicative and associative.

Solution

An operator o is commutative if X o Y = Y o X

Here x*y = xy + 2ax + 2y

=> y*x = yx + 2ay + 2x

now x*y = y*x

=> xy + 2ax + 2y = yx + 2ay + 2x

=> 2ax + 2y = 2ay + 2x

=> Equating the coefficients of x and y, we get 2a = 2

=> a = 1

Therefore a is 1.

( b has not been in the expression)