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Is there notation for "for all unique..."? For instance, suppose you wanted to consider all distinct $x,y$ in some set $S$. Would we type $\forall !x,y\in S$?

Can we use "distinct" and "unique" interchangeably in this sense?

I've never seen notation for this and a google search/SE search did not find anything.

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    No, you can't use them interchangably... "distinct" and "unique" are distinct words. – Zev Chonoles Sep 23 '13 at 05:33
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    "For all unique..." doesn't make sense, so there's no point to choosing a notation for it. – Zev Chonoles Sep 23 '13 at 05:35
  • One should say not "for all unique" but "for all distinct." So there is no parallelism between the common $\exists!$ and your suggested $\forall !$. – André Nicolas Sep 23 '13 at 05:36
  • @ZevChonoles Perhaps you could explain why it does not make sense in an answer? –  Sep 23 '13 at 05:36
  • Also, if anyone could explain how this is a duplicate question to that in the notice, I would be interested. I'm not suggesting that $\forall !$ would be the negation of $\exists !$. I'm not sure how my question is answered by the answers to the linked question. –  Sep 23 '13 at 05:47
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    I have seen $\forall x\neq y$ for this, but one has to be careful that there can be no doubt here that both $x$ and $y$ are new variables. – Jens Sep 23 '13 at 11:51
  • @Jens I can see how this would be taken to denote all $x$ that are not equal to a specific $y.$ Perhaps $\forall(x\neq y)$ would be better... –  Sep 23 '13 at 14:00
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    I would use $\forall x((x \ne y) \to \cdots)$. Clarity is more important than saving a few characters. – Jay Sep 23 '13 at 14:20

3 Answers3

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I'd be wary of using $\forall !$ as your notation for this. (See here for example.) The word 'unique' definitely doesn't mean what you want to say, though. 'Unique' means 'one', and 'distinct' means... exactly the opposite!

You'd be better off writing "for all distinct $x,y \in S$".

Working formally, if you were trying to show that a property $\phi(x,y)$ holds for distinct $x,y \in S$ you could write $$(\forall x,y \in S)(x \ne y \to \phi(x,y))$$ ...but outside of logic there isn't really much need for this kind of notation.

Community
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Clive Newstead
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  • You're explanation regarding "unique"vs."distinct" is very helpful! However, why do you say that there is no need for this kind of notation? I'm sure people have once said that about notation that has since been adopted into common practice... –  Sep 23 '13 at 05:58
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    The way I look at the purpose of mathematical notation, is that it serves to aid in the communication of mathematical ideas. Good mathematical writing will involve a balance between symbols and sentences. We adjust the lenses of symbols and words so that the mathematical ideas come into focus; overuse of either doesn't produce good mathematical pictures. I think Clive means that, at least in the context of communicating mathematical ideas, the need to use this notation does not necessarily show anything that simple words would not; hence we don't have much use for such symbols, (within maths). – Nirav Sep 23 '13 at 06:40
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The word "unique" means having a property that distinguishes it from all other things. In mathematics, this is meaningless if not followed by "such that" and the precise property that distinguishes it from everything else. Every $x$ is the unique $y$ such that $y=x$, so being unique without qualification is void of meaning.

The one useful use of "unique" in mathematics is therefore in the context "the is a unique $x$ (in a given set $X$) such that the property $P(x)$ holds", which translated into a logical formula gives $\exists x\in X:\bigl(P(x)\land \forall y\in X:P(y)\to x=y\bigr)$.

The word "distinct" applies to a a family of variables (often just two or three): the variables $x_i$ for $i$ in some index set $I$ are all distinct whenever $x_i=x_j$ implies $i=j$ (the map $I\to X:i\mapsto x_i$ is injective). It is not attached to universal quantification (saying "for all distinct $x$" is void of meaning), though the variables $x_i$ might have been introduced by quantification (either universal or existential, both are quite reasonable).

So there is nothing that links "unique" with "distinct", except that both are non-properties: they are void of meaning when applied (without qualification) to a single variable. I think what may be confusing you is the horrible abuse of "unique" that is sometimes used in combinatorics (or probability): determine the number of unique configurations of some kind. I never know what "unique" is supposed to mean in this context. It might mean "distinct" in the sense that each configuration is counted only once, but that is rather silly since this is implicit in counting things correctly. Sometimes "number of distinct" such and such may be used to indicate not attaching a multiplicity which the reader might otherwise be inclined to do: the number of distinct roots of a polynomial, or eigenvalues of a matrix. However one would never say "number of unique" in this context. More likely the "number of unique configurations" means count only classes for some (not explicitly indicated) equivalence relation of configurations, such as orbits for a symmetry group acting on the set of configurations (seatings around a round table...).

In any case "for all unique $x,y,z$" is meaningless. One could say "for all distinct $x,y,z$", which is a slightly informal way of saying for all $x,y,z$ such that $x\neq y\neq z\neq x$ (don't forget the last inequality!).

Marc van Leeuwen
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Here is a definition of "unique" that I got by Googling:

being the only one of its kind; unlike anything else

Let $X$ be a set. Let $x$ be an element of $X$. Is $x$ unique? Of course, in an utterly useless sort of way: $x$ is unique because anything other than $x$ isn't $x$. I see the only interpretation of

For all unique $x,y\in X$, ...

as being

For all $x,y\in X$ such that $x$ and $y$ are unique, ...

or in other words

For all $x,y\in X$, ...

Now, if you want to use the word "unique" to mean something other than what everyone else uses it to mean, e.g. "distinct", you're welcome to do so, but you should expect there to be confusion.

Zev Chonoles
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    "...to say of two things that they are identical is nonsense..." That part of the Wittgenstein quote is incorrect in this context. We use the word "distinct" because two things can be identical. – Jonas Meyer Sep 23 '13 at 06:04
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    @Jonas: Indeed - it makes perfect sense, it's just false. I just wanted to include this quote since it captures the basic point. – Zev Chonoles Sep 23 '13 at 06:07
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    What is false? What I mean is that when we speak of two things in many contexts in mathematics, it can be true that they are identical. Hence the need to say explicitly when we mean to speak of two different (or distinct) things. – Jonas Meyer Sep 23 '13 at 06:10
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    @Jonas: The quote only makes sense if you read the phrase "two things" to mean "two distinct things" (which is its colloquial meaning). This doesn't seem too unreasonable; since after all the quote uses the colloquial meaning of "nonsense". – Zev Chonoles Sep 23 '13 at 06:12
  • I've removed the quote, since I feel like this discussion of what Wittgenstein meant isn't helping my actual point. – Zev Chonoles Sep 23 '13 at 06:28
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    @JonasMeyer: the precise mathematical meaning of "two things" is "two distinct things", otherwise the "two" is wrong. One should however distinguish between the things and the names by which we designate them: $x$ and $y$ are distinct names, that could well designate one same object. So saying "let $x,y$ be elements of $X$" is not the same as saying "let $x,y$ be two elements of $X$" (which implicitly requires $x\neq y$). Of course one can find the formally incorrect phrase "the two values must be equal" in millions of places. – Marc van Leeuwen Sep 23 '13 at 07:16
  • @ZevChonoles Thank you for the clarification! –  Sep 23 '13 at 08:14
  • I'm more interested in where you found the definition than I am in how you found it. Please don't say you found something "by googling". That doesn't tell me much. – TRiG Sep 23 '13 at 10:13
  • @TRiG: Except Google *is* where I found the definition. If there were any other relevant information I would have provided it. – Zev Chonoles Sep 23 '13 at 12:50
  • Zev, @Marc: Thanks, I agree, I was mistaken. Sorry for the noise. Zev, I'd be happy to delete my comments if you want, to remove the clutter (and maybe reinstate the interesting quote). (I only hesitate to delete because it would render the responses confusing.) – Jonas Meyer Sep 24 '13 at 02:27