function

A function \( f(x) \) is a set \( S \) of ordered pairs that map the first value of the ordered pair to the second value in the ordered pair, where the first value may not have duplicates in the set \(S\). The map from the first value to the second value has the notation \(f(x) = y \) for some x and y, where \(f\) is the mapping. Note that we can also define rules for \(f\) and do not therefore have to explicitly define all the mappings:

\begin{align*} S = \{(x, y): x^{2} = y, x, y \in \mathbb{R} \} \end{align*}

Which is an example of a parabolic function. \(x\) and \(y\) can both conceptually be any object, but usually they are mathematical objects. Some examples of such objects include tensors and scalars.

However, we must find a way to define what an ordered pair is. Sets have no order by default, so we need to add order by doing the following:

\begin{align*} (x_{0}, y_{0}) := \{x_{0}, \{x_{0}, y_{0}\}\} \end{align*}

Where the element that is not explicitly a set gives us the definition of the first element.

Let \((S, \circ)\) define a group where \(S\) is the set of all functions, and \(\circ\) is the composition binary operator. Then \(f(x) = x\) is the identity element, and an inverse of a function is defined as \( (f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x \).