How to symbolically write "there exists at most"

Question

For example, an injective function can be defined as

$\forall y \in Y$ there exists at most one $x \, : \, \forall f(x) = y$

How could I represent the bold part of the above expression? For "there exists at least one" we have $\exists$, for "there exists a unique" we have $\exists!$. Is there a symbolic notation for "...at most one"? The classic definition of injective, i.e., $\forall \, a,b\in X, \, f(a) = f(b) \rightarrow a = b$ seems to side-step this, though.

Answer 1

If you will be using it repeatedly, you could invent a notation... $$ \forall y \in Y\;\exists^{\le 1} x $$ Of course you will have to define it, say like this $$ \exists^{\le 1} x\;\;\Phi(x)\qquad\text{means}\qquad \forall x_1 \forall x_2 \big(\Phi(x_1) \wedge \Phi(x_2) \longrightarrow x_1=x_2\big) $$

Answer 2

I have never heard of such a symbol, but if you really want to you can write $\exists !\, x \lor \nexists\, x, \,\dots $

However, this is not easily readable and I would not advise its use at all.

Answer 3

For example, an injective function can be defined as

$\forall y {\in} Y$ there exists at most one $x : \, f(x) = y$

How could I represent the bold part of the above expression? For "there exists at least one" we have $\exists$, for "there exists a unique" we have $\exists!$. Is there a symbolic notation for "...at most one"? The classic definition of injective, i.e., $\forall \, > {a,b}\in X, \, f(a) = f(b) \rightarrow a = b$ seems to side-step this, though.

If you accept a string of symbols in lieu, then how about:

$$\forall y{\in}Y \; \exists p\;\forall x{\in}X\:\Big(f(x)=y \implies p=x\Big). $$