Cohen forcing question

Question

Suppose $M$ is a countable transitive model of ZFC and $(x, y, z)$ is Cohen generic point in $\mathbb{R}^3$ over $M$: This means that for every open dense set $U \subseteq \mathbb{R}^3$ in $M$, $(x, y, z) \in U$. I would like to know if $(x+y, x+z, y+z)$ is also Cohen generic point in $\mathbb{R}^3$ over $M$. I tried using product forcing lemma but couldn't show this. Neither could I get a counterexample. Thanks for any ideas!

Answer 1

The function $\vec x\mapsto A\vec x$ defined by $(x,y,z)\mapsto(x+y,x+z,y+z)$ is an invertible linear operator (with inverse $(a,b,c)\mapsto\frac12 (a+b-c,a-b+c,-a+b+c)$) definable in the ground model $V$. Thus it maps dense open sets to and from dense open sets in the ground model, so for any dense open $U$ in $V$, $A(x,y,z)\in U\iff (x,y,z)\in A^{-1}U$, which is dense open and in $V$. Hence $A(x,y,z)$ is a generic point.