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:PROPERTIES:
:ID: b1f9aa55-5f1e-4865-8118-43e5e5dc7752
:END:
#+title: function
#+author: Preston Pan
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* Definition
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.
* ordered pair
:PROPERTIES:
:ID: 1b1b522e-d4de-4832-9ca4-c6d1cfee27e6
:END:
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.
* Function Group
Let \((S, \circ)\) define a [[id:ba7b95b0-0ce6-4b33-9a79-5e5fddaea710][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 \( (f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x \).
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