How to read parens How do you read the parentheses in this proposition? That is, what do you say in English when reading from the end of a parenthesis to the next? Are the parens simply read as "such that"?
$
(
∀
x
∈
Z
) (
∃
y
∈
Z
)
x < y\\(
∃
y
∈
Z
) (
∀
x
∈
Z
)
x < y$
 A: The parentheses are just read "as such". Note, though, that when reading things aloud, it is sometimes hard to get verbalize the details of the statements.
So $$(\forall x\in Z)(\exists y\in Z) x<y$$ would (could be) read
For all $x$ in $Z$ there exists a $y$ in $Z$ such that $x$ is less than $y$.
Note that when you have more complicated expressions, reading things out becomes more difficult. If you for example have $(x+y)z$, then this is $x$ plus $y$ times $z$. The problem is that it isn't clear that there are parentheses around the sum. In this case you could maybe say: $z$ times the sum of $x$ and $y$. In other words, you sometimes have to be a bit "creative".
A: i havn't encountered using brackets like this in such a statement, so top of the head i would write and read it like this:
1) for all $x$ in$Z$ there exists a $y$ in $Z$ such that $y<x$
or : $\forall x\in Z \exists y\in Z:x<y$
2) there exists a $y$ in $Z$ such that for all $x$ in $Z$: $x<y$ holds.
or: $\exists y\in Z\forall x\in Z:x<y$
if this is not what you are looking for, please clarify(you or someone who is familiar with it) the use of the brackets to me.
A: The purpose of the parentheses here is legibility. The convention is of long standing, and I consider it best practice. It goes naturally with enclosing in brackets the statement to which the quantifiers apply: for example, $$(\forall\varepsilon\in\Bbb R_{>0})(\forall x\in\Bbb R)(\exists\delta\in\Bbb R_{>0})(\forall y\in\Bbb R)[|y-x|<\delta\Rightarrow |f(y)-f(x)|<\varepsilon].$$
