# Notation

2015-08-26

## Building sets

$\mathcal S \times \mathcal S = \mathcal S^2$

## Notation

### Binding operators

Binding operators have the generic notation

$$\O[\mathcal S] x$$

where $\mathrm{O}$ denotes the operator to be preformed, $\mathcal S$ is a set from which elements $x$ will be taken. The operator is right-associative and hence applied on the expression following it, which may include references to $x$. The resulting expression will not depend on $x$, it is bound by the operator.

Many operators are defined by a fold, for example the summation operator can be written out using binary addition and zero as $\mathrm{\Sigma }^{n}_{ℕ} f(n) = 0 + f(0) + f(1) + \cdots$. The zero in front is important, otherwise the result would be undefined if an empty set was given. The operators defined in this way are:

Operator Join Base
$\mathrm \forall$ $\wedge$ $⊤$
$\mathrm \exists$ $\lor$ $⊥$
$\mathrm \cup$ $\cup$ $\varnothing$
$\mathrm \cap$ $\cap$ $\mathcal U$
$\mathrm \Sigma$ $+$ $0$
$\mathrm \Pi$ $\;\cdot \;$ $1$

The expression is a function with $\mathcal S$ as its domain and a the same range as the operator expression.

### Exceptional binding operators

Operator Interpretation
$\mathrm \nexists$ not exists
$\mathrm \exists !$ unique
$\mathrm \exists ?$ an element such that
$\mathrm \exists ‽$ the unique element such that
$\mathrm \vert$ substitute
$\mathrm S$ set builder
$\mathrm L$ limit
$\mathrm \int$ integral
$\mathrm max$ maximum
$\mathrm argmax$ argument of the maximum

The substitute operator takes $\mathcal S$ as a singleton set, in fact, we can forgo the set notation $\left\{ x \right\}$ and write x directly. The set builder takes a boolean expressions, but gives a subset of $\mathcal S$ as result. In limits the set $\mathcal S$ is a directed set.

$$\L[ℕ]n \frac{1}{n} = 0$$

The limit is defined using a generalization of the $(\epsilon , \delta )$ definition:

$$\L[\mathcal S]x f(x) = l \iff \mathrm{\forall }^{U}_{\mathcal U(l)} \mathrm{\exists }^{\alpha }_{\mathcal S} \mathrm{\forall }^{\beta }_{\mathcal S \geq \alpha } f(\beta ) ∈ U$$

$$\L[\mathcal S]x = \mathrm{\exists ‽}^{l}_{} \mathrm{\forall }^{\epsilon }_{\mathcal U(l)} \mathrm{\exists }^{\delta }_{\mathcal S} \mathrm{\forall }^{x}_{\mathcal S \geq \delta } \epsilon ∋$$

where $\mathcal U(l)$ is the set of all neighborhoods around $l$ and $\mathcal S \geq \alpha$ is defined as $\S[\mathcal S] \beta \beta \geq \alpha$.

A Riemann integral is defined as

$$\mathrm{\int }^{x}_{[0,1]} f(x) = \L[ℕ]n \frac 1 n \mathrm{\Sigma }^{i}_{[0,n-1]} f\left({\frac i n}\right)$$

Using the substitute operator this can be written concisely as

$$\mathrm{\int }^{x}_{[0,1]} = \L[ℕ]n \frac 1 n \mathrm{\Sigma }^{i}_{[0,n-1]} \mathrm{\vert}^{x}_{\frac i n}$$

The fundamental theorem of calculus can be written as:

$\mathrm{\int }^{x}_{[a,b]} = \left({ \mathrm{\vert}^{x}_{b} - \mathrm{\vert}^{x}_{a} }\right) \operatorname*{D}\nolimits_{x}^{-1}\;$

### Non binding operators

\begin{aligned} &\mathrm{\exists }^{x}_{\mathcal S} P(x) &&\iff & \S[\mathcal S]x P(x) &\neq \varnothing \\ &\mathrm{\exists !}^{x}_{\mathcal S} P(x) &&\iff & \left\vert{\S[\mathcal S]x P(x)}\right\vert &= 1 \end{aligned}

$r(\mathcal S)$ arbitrary element from $\mathcal S$.

## Integral transforms

$\mathrm e$ $\mathrm i$ $\mathrm \pi$.

$\hat f(\omega ) = \mathrm{\int }^{t}_{ℝ} \mathrm e^{-2 \mathrm \pi \mathrm i\omega t} f(t)$

$\mathcal F_\omega ^t = \mathrm{\int }^{t}_{ℝ} \mathrm e^{-2 \mathrm \pi \mathrm i\omega t}$

$\mathcal L_s^t = \mathrm{\int }^{t}_{[0,\infty )} \mathrm e^{-s t}$

$\hat f(\omega ) = \mathcal F_\omega ^t f(t)$

https://en.wikipedia.org/wiki/Integral_transform

Generalized:

$\mathcal T(K, \mathcal D)_u^t = \mathrm{\int }^{t}_{\mathcal D} K(u, t)$

## Extended real numbers

The extended real numbers,

$\overline ℝ = \left\{ -\infty , +\infty \right\} \cup ℝ$

## Set building

Given a proposition $P : \mathcal S \rightarrow \mathbb{B}$ we can the set builder operator is defined by:

$$\mathcal Q = \S[\mathcal S] x P(x) \iff \mathrm{\forall }^{x}_{\mathcal S} x ∈ \mathcal Q \leftrightarrow P (x)$$

$\mathrm{\forall }^{f}_{ℝ\rightarrow ℝ} \mathrm{\exists }^{y}_{ℝ} \mathrm{\int }^{x}_{[0,2,4,\ldots ]} f(x) = \mathrm{\Sigma }^{x}_{ℝ} f(x)$

$\mathrm{\int }^{x}_{} f(x) =$

$123 \operatorname*{∂}\nolimits_{z}^{}\; 2 \operatorname*{D}\nolimits_{y}^{}\; \operatorname*{\delta }\nolimits_{x}^{}\; f(x)$

The Laplacian operator:

For some rectalinear basis $\mathcal B$:

\begin{aligned} \vec ∇ &= \mathrm{\Sigma }^{\vec x}_{\mathcal B} \vec x \operatorname*{∂}\nolimits_{\vec x}^{}\; \\ \Delta &= {\vec ∇}^2 = \mathrm{\Sigma }^{\vec x}_{\mathcal B} \operatorname*{∂}\nolimits_{\vec x}^{2}\; \end{aligned}