Interpreting statement notation

Question

I have statement

$$ \forall x \in \mathbb{Z} : \exists y \in \mathbb{Z} : xy=1 $$

I would translate this into English as:

for all x and for some y integers it is true that:

$$ xy=1 $$

Problem is i don't understand $:$ in this notation. To my understanding if we have statement it is in form $\text{statement}:\text{condition}$ and for sets is used $\{ \text{statement}:\text{condition} \}$ (same thing with $\{\}$). When you have mulitple $:$ in same sentence i don't know how which are "statements" and which are "conditions". With this exmaple it is easy to determine which is "statement" and which is "condition" but there might exists some cases which this isnt so clear ?

Answer 1

I'd rather call the parts quantification and statement. Example: $$\underbrace{\exists x \in X}_{\text{quantification}} : \underbrace{P(x,y)}_{\text{statement}}$$

Note that the full formula above is also a statement and can be used with another quantification: $$\underbrace{\forall y \in Y}_{\text{quantification}} : \underbrace{\big( \underbrace{\exists x \in X}_{\text{quantification}} : \underbrace{P(x,y)}_{\text{statement}} \big)}_{\text{statement}}$$

There is no need for the extra pair of parentheses though so we just write $$\forall y \in Y : \exists x \in X : P(x,y)$$ since the interpretation of this as $\big( \forall y \in Y : \exists x \in X \big) : P(x,y)$ doesn't make sense.

Note that often the colon is not used and you might meet $\forall y \in Y \ \exists x \in X \ P(x,y)$. Others put parentheses around the quantifications and thus write $(\forall y \in Y) \ (\exists x \in X) \ P(x,y)$.

Answer 2

Your interpretation of the statement is correct, though precisely it means for all $x$ there exists a $y$ such that $xy=1$. That is, there is at least 1 $y$ for which this statement is true for all $x$.

As others have mentioned, the colons separate the statement from the conditions. But there are two colons! you might object. But however you split the statement, the meaning is the same. Usually the colon can be interpreted as such that, but it can also be interpreted simply as a comma, it depends on the context. These two statements are the same:

Statement 1: [for all $x$, there exists a $y$ such that]:$\implies$[$xy=1$].

Statement 2: [for all $x$]:$\implies$[there exists a $y$ such that $xy=1$].

No matter where you split the statement from the condition, the logical statement is the same.