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I find the word "iff" for "if and only if" quite helpful for brief statements, but is there a similar one meaning "one and only one"?

edit In light of the ambiguities some of the answers so far hint at, here's an example I'd like to shorten:

Of the statements x, y, z, one and only one statement is true

should become

xorne of the statements x, y, z is true

where xorne is the sought word.

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9 Answers

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I have never come across such a word. My best ideas for replacements are: Precisely one, Exactly one, just one.

Sidenote: Some Authors prefers "if and only if" over "iff", since it can be easy to forget to show both ways. So maybe it is a good thing that there is no shorthand, if indeed there is none.

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John H. Conway has said that whenever he writes "iff", he finds himself compelled to also write "thenn" (ie, "then and only then!") for parallel emphasis in both parts of a statement. (It's only fair, after all, given that the statement asserts the parts' logical equivalence.)

I see no reason that we can't follow his lead (as well as consonant-doubling precedent) and introduce, say, "onne" into the mathematical vernacular.

Edit. According to @AndreasBlass, Conway (unsurprisingly) beat me to it.

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Edit. User @Someone else posted a sourced answer linking Conway to "onne", then deleted that answer for being a "duplicate" of mine. OP and I find the reference(s) helpful, so I'll quote the post here. (If @Someone has an objection ---or if this is a violation of some StackExchange policy--- I can delete it.)

J H Conway of Princeton introduced "onne" for "one and only one". (Reference: Margie Hale: "Essentials of Mathematics: Introduction to Theory, Proof, and the Professional Culture")

I have seen this a few times in books and scholarly papers, although it is obviously much less common than "iff".

In a follow-up comment, @Someone wrote:

Here's a second (although possibly not very reliable) source for the same thing - MathForum post.

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You can see it written symbolicallly: $$ \exists ! $$ So like this:

$(\exists! x \in \mathbb N)( x^2=4)$

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A unique. ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$

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"$\exists$ a unique" would be the shortest common way to say what you're asking for.

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Although there are many answers at this point, I'm compelled to answer because the original question did not use a quantifier, so depending on the logic it may be difficult to use $\exists!$ directly. For example, the statement "exactly one of the sets $x,y,z$ is blue" can be written "$\exists!n\in\{x,y,z\}\, n$ is blue", but "exactly one of the following propositions is true: the moon is made of cheese, I know how to swim, there is an elephant in the White House" is much harder to write in this manner, because there are no sets available to quantify over (propositions can not necessarily be treated as sets; the difference is what lead to Russell's paradox).

If $P,Q,R$ are propositional variables, then "exactly one of $P,Q,R$ is true" can be faithfully represented as:

$$(P\land\lnot Q\land\lnot R)\lor(\lnot P\land Q\land \lnot R)\lor(\lnot P\land\lnot Q\land R)$$ $$(P\oplus Q\oplus R)\land\lnot(P\land Q\land R)$$ $$((P\oplus Q)\land\lnot R)\lor(\lnot P\land\lnot Q\land R)$$

where $P\oplus Q=(P\land\lnot Q)\lor (\lnot P\land Q)$ is the exclusive or.

The third example actually generalizes well; if $\phi_k^n$ means "exactly $k$ of $P_1,P_2,\dots,P_n$ is true" then these formulas can be defined inductively as

$$\phi_0^1=\lnot P_1\quad\phi_1^1=P_1\qquad\phi_0^n=\phi_0^{n-1}\land\lnot P_n\quad\phi_n^n=\phi_{n-1}^{n-1}\land P_n\qquad$$ $$\phi_k^n=(\phi_{k-1}^{n-1}\land P_n)\lor (\phi_k^{n-1}\land\lnot P_n)$$

although the length of the formula is still quite long, ${n+2\choose k+1}-2$ in general, and I don't know if there is an asymptotically shorter formula.

One last special case: The number one case (in fact, the only case) where I've seen this propositional "exactly one" occur in math, and which may be the prototype from which you abstracted this question, is in the statement of the trichotomy law for real numbers:

If $x,y$ are real numbers, then exactly one of $x<y$, $x=y$, $x>y$ is true.

In this particular case (in the presence of antisymmetry), it turns out that you can rewrite this law into a simpler one using only the biconditional:

If $x,y$ are real numbers, then $\lnot\, x<y\leftrightarrow(x=y\lor y<x)$.

It is a good exercise to show that this axiom implies both trichotomy and antisymmetry.

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"xorne" would usually be written as "precisely one" or "exactly one." I don't think there is an accepted abbreviation; if you really want to find something, your best bet is probably to research the family of functions $f_n : \mathbb{B}^n \rightarrow \mathbb{B}$ defined by asserting that $f_n(x)=1$ iff there is a unique $i \in \{1,\ldots,n\}$ such that $x_i=1$.

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$\exists !$ won't really work by itself as the OP is potentially restricting to a subset of a larger space, i.e. there could be many objects that satisfy the property in question, but only one out of a specific collection which satisfies the property: $\exists ! x\in A\subset E$ such that $\Phi(x)$, but $\exists y\in E$, such that $y\neq x$, and $\Phi(y)$.

So we need something more along the lines of $$``\text{there is only one element of}" = \exists ! (\text{fill-in-the-blank})\in.$$

Any word would work, you just have to define it as there isn't any standard. "xorne" isn't quite as nice as "iff" or "thenn" as the latter are very natural sounding in speech, and are almost a natural slur of the statements they substitute for. In light of that, I would suggest "olnel" for "only one element of" (pronounced \ōl'nel\).

Olnel $A$ $\Phi$ = "only one element of set $A$ satisfies statement $\Phi$."

Edit: On second thought... How about "onelf"? Pronounced one elf. Or " alonelf" pronounced a lone elf for "a lone element of".

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Ok, for the sake of it, I propose

xorne

which portmanteaus xor and one as an actual answer.

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