$\forall x,y,z \in \mathbb{R} (x\cdot 0 + y \cdot 0 + z \cdot 0=0) \to (x=y=z=0)$

Question

$\forall x,y,z \in \mathbb{R} (x\cdot 0 + y \cdot 0 + z \cdot 0=0) \to (x=y=z=0)$?? If is true, why? Thanks in advance!

Answer 1

If $+$ and $\cdot$ mean usual addition and multiplication of real numbers, and if $0$ means the particular zero element of the reals, then the predicate $P(x,y,z)=0$ holds for any triple $(x,y,z).$ In that sense, this predicate does not mean the same as $x=y=z=0$, as the latter predicate is only true at $(0,0,0)$ and yours is true at any triple $(x,y,z).$

Edit: I should have used predicate notation for both. That is, $P(x,y,z)$ is the predicate $x\cdot 0 +y \cdot 0 + z \cdot 0=0$, and it happens that $P(x,y,z)$ holds for any choice of the variables $x,y,z$. And we can define $Q(x,y,z)$ to be the predicate $x=y=z=0$, so that only $Q(0,0,0)$ is true, while $Q$ is false for other choices like $Q(1,3,2).$ The relation is that if $Q$ holds then $P$ also holds, but not conversely. Note that the same effect as $P$ could be obtained by defining $P$ to be $x=x.$ This is also true for any triple $x,y,z.$