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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$

George Newton
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  • I read it like this: for every $x\in Z$ there exists $y\in Z$ such that $x<y$. – Ludolila Jan 30 '14 at 13:14
  • Those symbols do not function as parentheses–this is just a notation convention for the quantifiers, rather than syntax changing the order of evaluation. Similar situation happens with functions, e.g. $f(x)$, where we don't read symbols "$($" and "$)$". You could think that in this case "forall" and "exists" take care of the first and "such that" handles the second. Anyway, natural language is not suited for describing formulas which often have more complicated structure than linear (e.g. fractions, matrices or, for complicated cases, expression trees). – dtldarek Jan 30 '14 at 13:37

3 Answers3

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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".

Thomas
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  • so the brackets are really redundant and do not impose a further structure on the logical argument? is this a common way of writing in some areas of math, since i've never seen this before. only without brackets. – quis23 Jan 30 '14 at 13:23
  • @quis23: In your specific examples the parentheses are redundant. However, as I usually tell my students, if you have doubts, then just put parentheses. – Thomas Jan 30 '14 at 13:28
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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.

quis23
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  • This is a one of many conventions on quantifier usage and syntax. Some write $\forall x:\ P$, some write $\forall x(P)$ and some write $(\forall x)P$ (there are some more examples here). – dtldarek Jan 30 '14 at 13:43
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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, $$(\forall\varepsilon\in\Bbb R_{>0})(\forall x\in\Bbb R)(\exists\delta\in\Bbb R_{>0})(\forall y\in\Bbb R)[|y-x|<\delta\Rightarrow |f(y)-f(x)|<\varepsilon].$$

John Bentin
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