Function notation with set of mappings valid or not? I have never seen one

Question

I have seen notations in maths scripts similar to the following:

$f: \{1,2\} \rightarrow \{3, 4\}$

$1 \mapsto 3$

$2 \mapsto 4$

but not once compact mappings similar to the relation in the set notation {(1,3), (2,4)}

Would you say the following definition is ambiguous or even incorrect?

$f: \{1,2\} \rightarrow \{3, 4\}, \{1 \mapsto 3, 2 \mapsto 4\}$

Answer 1

I am able to know exactly what you mean here. So I don't think it's ambiguous or incorrect...

Of course, you might instead write: $f(1)=3$ and $f(2)=4$...

Answer 2

I don't see the point. Instead of introducing the new notation $$ f: \{1,2\} \rightarrow \{3, 4\}, \{1 \mapsto 3, 2 \mapsto 4\}$$ why not simply use the shorter, already established notation $$ f: \{1,2\} \rightarrow \{3, 4\}, \ 1 \mapsto 3, \ 2 \mapsto 4 \ \ ?$$

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You could also use $$ f = \{1 \mapsto 3, 2 \mapsto 4 \}$$ which is a perfectly reasonable notation as far as I'm concerned.