In each of the following cases prove that the given function

In each of the following cases, prove that the given function is a homomorphism and describe its image and kernel. (a) The function f : R times R rightarrow R defined by f((x, y)) = 2x - y.

Solution

A homomorphism is a structure-preserving map between two algebraic structures. Here, the function f : R x R R Then f is a homomorphism of groups, since it preserves addition.

image( f ) = {f(x, y): x , y R} = { z R : z = f(x, y), for some x, y R} = { 2x –y : for some x, y R}

The kernel of the function f is   the set of all zeros of the function (i.e., the solutions of the equation f (x) = 0 i.e. { x, y R : 2x – y = 0}