# Predicate logic notation: where to put the parentheses, etc.

My math professor tends to write $\exists x\in\mathbf{X} \ni P(x)$. Is this a correct use of the such that symbol $\ni$? If not, what is the use of that symbol? Isn't it better to write $\exists x\in\mathbf{X} (P(x))$ assuming $P(x)$ is a complex proposition? Also, while we're at it, is $\forall x\in\mathbf{X} : P(x)$ an acceptable notation or should it really be $\forall x\in\mathbf{X} (P(x))$ for precision?

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I'm guessing he means it as "such that." Often we just use a colon: $\exists x\in\mathbf{X}: P(x)$. You could put parentheses around it, as you do. Incidentally, even that is shorthand, since what it really means is: $\exists x(x\in\mathbf{X} \text{ and } P(x))$. –  Thomas Andrews Jan 27 '12 at 13:54
This is semi-formal notation, at some distance from the standard ways of expressing the idea in a formal language. –  André Nicolas Jan 27 '12 at 15:15
I would describe $\exists x\in\mathbf{X}\ni P(x)$, $\exists x\in\mathbf{X}: P(x)$, and $\exists x\in\mathbf{X}$ s.t. $P(x)$ as shorthand for an English sentence, There is an x in ... . $\exists x\in\mathbf{X}\big(P(x)\big)$ and $(\exists x\in\mathbf{X}) P(x)$, on the other hand, are expressions in slightly different formal languages, abbreviating the even more formal $\exists x\big(x\in\mathbf{X}\land P(x)\big)$ or $(\exists x)\big(x\in\mathbf{X}\land P(x)\big)$. –  Brian M. Scott Jan 27 '12 at 15:29

The backwards epsilon means "such that", but in this context it's slightly bizarre since the usual set membership symbol appears symmetrically before the $\mathbf{X}$ and of course means something quite different. After all, if $P$ were a function symbol rather than a relation symbol then we might parse the statement quite differently, as $\exists{x} (x \in \mathbf{X} \wedge P(x) \in \mathbf{X})$.
We can't see inside your professor's mind, but with usual usage in mind, your expansion is correct: it abbreviates $\exists{x \in \mathbf{X}} (P(x))$, which is in turn an abbreviation for $\exists{x} (x \in \mathbf{X} \wedge P(x))$.
Universal quantifiers expand similarly: $\forall{y \in Y} \; \varphi(y)$ is short for $\forall{y} (y \in Y \rightarrow \varphi(y))$. Both colons and full stops (periods) are acceptable separators: $\exists{y \in Y} : \varphi(y)$ and $\forall{z \in Z} \; . \; \psi(z)$ are reasonably common.