Restate a logical claim using logical symbols Proposition: Strictly between any two distinct rational numbers lies another rational number.
How may I present this statement using logical symbols?
My answer: $\forall x, y \in {\mathbb{Q}}. \space x \neq y \implies \exists z \in {\mathbb{Q}}. \space (x < z < y) \vee (y < z < x)$
I think there is some problem with my statement because I suspect $\forall x, y \in {\mathbb{Q}}. \space x \neq y$ doesn't make much sense.
Could you please shed some light on how I may best write this statement? Thanks.
 A: You've done fine, except for an important matter of syntax: You need to bracket the entire implication in which $x$ and $y$ appear.
Without doing so, the appearance of $x, y$ in the consequent are unbound, i.e., represent free variables, and hence, have no relation to the bound $x, y$.  You also need to bracket the entire disjunction following $\exists z...$, for the same reason, otherwise the second appearance of $z$ is  unbound and has no relation to the quantified variable $z$.
This can be remedied by writing your symbolic statement as follows:
$$\forall x, y \in {\mathbb{Q}} \space \Big(x \neq y \implies \exists z \in {\mathbb{Q}} \space [(x < z < y) \vee (y < z < x)]\Big)$$
A: It makes sense to me. When you write
$$
\forall x, y \in \mathbb{Q}, x \neq y \Rightarrow \ldots,
$$
it means that if it happens that the $x, y$ you chose are not equal, then they must satisfy some property (here the property that another rational number lies between them). If it happens that $x = y$ then the implication is automatically true since the premise is false.
