TITLE: What does this combination of symbols mean? $\exists !$
QUESTION [5 upvotes]: I just want to know what this combination of symbols means:
$\exists !$
I know ∃ means 'there exists', but what does it mean when it is paired with a '!'? I have written down 'there exists unique" but I am not 100% sure this is correct?

REPLY [8 votes]: $\exists ! x:$ "There exists a unique x..."
To use Carlos' example: $$\exists ! m \in \mathbb R,\,\forall a \in \mathbb R( a\cdot m = a)$$
This is equivalent to the statement:
$$\exists m \in \mathbb R, \forall a \in \mathbb R\Big(a\cdot m = a \land \forall y \in \mathbb R((a \cdot y = a) \implies y = m)\Big)$$
As you can see, the uniqueness quantifier greatly simplifies the logical statement.