# Notation for any member of a set

Is there any notation that can represent any member of a particular set? For example, how could I say that I want any real number, or any imaginary number?

I do know there is a symbol to denote "for all", $\forall \Re$, but I don't want that. I want to express that for example that a operation will return any member of the real set.

Is there such a notation?

• If the operation $f$ takes elements of set $A$ as its inputs and always returns an elements of set $B$ as its outputs, one usually writes $f:A\to B$ to indicate this. So if $a\in A$, then $f(a)\in B$ (whatever it is). Of course, this doesn't specify the precise rule of assignment, so it is assumed that you have some particular rule in mind. – MPW Sep 7 '16 at 13:18
• @MPW I know that too. What i want is for example when defining a matrix. if $i \geq j$, $a_{ij}=$ any member of $\Re$. Else $a_{ij}= 0$. It's strange. My teacher simply did $a_{ij} = \forall$ but I'm pretty sure that's not the right notation. – ChemiCalChems Sep 7 '16 at 13:21
• Can't you just use $a_{i j}\in \mathbb{R}$ then? – simonzack Sep 7 '16 at 13:36
• @simonzack Didn't think about that. Thanks. – ChemiCalChems Sep 7 '16 at 14:53

## 1 Answer

This concept has been used extensively before, I believe it was introduced by Hilbert as the ε-operator in a logical calculus called the ε-calculus. The syntax is $ε\ x\ A(x)$ or $ε\ x.A(x)$ which mean any $x$ satisfying the property $A$. In your case the property you are using is the membership in a set, so you would write $ε\ x.x \in A$ (though I'm not sure if this was allowed in the original calculus). The ε-operator is also used in some formal proof systems, such as HOL (Higher Order Logic).

See also the entry in the Stanford Encyclopedia of Philosophy.

• So if I understand correctly $\varepsilon x, x \in \Re$ would mean any x such that it belongs to the real numbers? That's nice! – ChemiCalChems Sep 7 '16 at 20:21
• yes, that's right. Also, the quantifiers $\forall$ and $\exists$ can be defined in terms of the ε-operator. – fairflow Sep 8 '16 at 6:58
• If you wanted to ensure that your matrix entries were positive on and above the diagonal, for example, you could use $ε\ x. x > 0$ for each element. In some versions of the ε-calculus, this would ensure all entries are equal (axiom of universal choice), in others this would not be guaranteed. – fairflow Sep 8 '16 at 7:02
• correction: since $z=z$ for every $z$, $ε\,x.x>0=ε\,x.x>0$ so all your entries would be the same; it is only if you used another relation equivalent but not equal to $>$ that you might not have equal elements. – fairflow Sep 8 '16 at 9:18
• @ChemiCalChems if you like my answer, you may consider accepting it if no other answers are forthcoming. – fairflow Sep 8 '16 at 9:20