# 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

## 1 Answer

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.

The key to using abbreviations effectively is to employ them to increase clarity. They increase brevity, but if using one might be ambiguous, go for the expanded version instead. I recommend reading things like Knuth on mathematical writing for further explanations of good practice in mathematical writing.

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