Do formal definitions of functions have to exclude one-many mapping?

Question

A formal definition of a bijection, for example, might be:

Let $f$ be a function whose domain is set $A$ and whose range is set $B$. $A$ function $f$ is bijective iff for every $b$ in $B$ there exists exactly one $a$ in $A$ such that $f(a)=b$.

This does not seem to exclude one-many mapping. Is this simply assumed, on the grounds that a function by definition cannot map one-many?

"Yes, it is" is the answer to "Is this simply assumed [...]?" — Sassatelli Giulio, Mar 06 '23 at 09:13
As a side note, if you consider your function with domain $A$ and final set $\mathcal{P}(B)$ (the power set of $B$), then it assigns to every $a\in A$ a subset of $B$. This is essentially a one-to-many function. — GReyes, Mar 06 '23 at 09:41

score 1 · Answer 1 · answered Mar 06 '23 at 12:01

1

Community wiki answer so the question can be marked as answered:

As noted in the comments, a function is by definition a relation in which every element of the domain is mapped to exactly one element of the range.

answered Mar 06 '23 at 12:01

joriki

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Do formal definitions of functions have to exclude one-many mapping?

1 Answers1