How do I define exactly what a function is?

Question

While it is easy to understand what a function is intuitively, I've been trying to wrap my head around how to precisely define what a function is using only mathematical notation. My attempt at this is below, but here is my preliminary understanding:

1. A function can have multiple inputs or parameters, but it generates a single output
2. Each output is unique for the input values provided.

Here is my attempt at a definition:

A relation $R \subset (D \times C)$ is a function if: $$ (\forall (d_1, c_1) \in R)(\forall (d_2, c_2) \in R)(d_1 = d_2 \rightarrow c_1 = c_2)$$

This definition should cover all functions, not just functions with one input, as $d_1$ and $d_2$ could be n-tuples that define the n inputs of a function, as every element of $R$ is actually an ordered pair $((x_1, x_2, ..., x_n),c)$

Does this look like a correct and precise definition? Or could it be written better? I couldn't find any formal definition of a function on the web, even on Wikipedia.

Finally, is it correct to say that all functions with n inputs are (n+1)-ary relations? Since $((x_1, x_2, ..., x_n),c)$ is the same as $(x_1, x_2, ..., x_n,c)$.

Thanks.

Answer 1

The usual way to define a function, you are correct, is by relations. However, your definition is not ok, because for example, an empty relation fits your definition.

The correct definition must say something like:

A relation $R\subset A\times B$ is a function if each element $a$ of $A$ is contained in exactly one relation $(a,b)\in R$

In pure terms, you can write this as

$$\forall a\in A\exists! b\in B: (a,b)\in R$$

Where the $\exists!$ quantifier means "exists precisely one". A longer version (with only $\exists$ and $\forall$) would be

$$\forall a\in A \exists b\in B:((a,b)\in R\land \forall b'\in B:(a,b')\in R\implies b=b').$$

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Answer to your other question about functions on multiple inputs:

No, functions are always binary relations. A function of "multiple (say, $n$ inputs)" is actuall a function whose domain is a cartesian product. So, for example, a function of $n$ real numbers is in fact a function whose domain is equal to $\mathbb R^3$.

This means that technically, we shouldn't write $f(x,y)=xy$, for example. We should write $f((x,y))=xy$, because $f$ takes only one input, and that input is a tuple of two numbers.

Answer 2

5xum is right -- your definition is not sufficient. However, you may be interested to know that your definition,

A relation $R \subset (D \times C)$ is a function if: $$ (\forall (d_1, c_1) \in R)(\forall (d_2, c_2) \in R)(d_1 = d_2 \rightarrow c_1 = c_2)$$

is something we care about in some branches of math! This is called a partial function from $D$ to $C$ (sometimes denoted $f: D \rightharpoonup C$).

Partial functions are especially used in the theory of recursive functions (or computable functions from $\mathbb{N}$ to $\mathbb{N}$).