I just want to know what this combination of symbols means:
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?
“There exist unique” $$\exists!m\in\mathbb R:\forall a\in\mathbb R:a\cdot m=a$$
In this example $\exists!$ means that $m$ is unique, that this number exist ($m=1$) but no other does have this property.
$\exists ! x:$ "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.
The symbol $\exists$ is the existential quantifier and with a $!$ in front of it, it means that there exists at most one element...