2015-08-26

Notation

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Building sets

$\set S × \set S = \set S^2$

Notation

Binding operators

Binding operators have the generic notation

$\O[\set S] x$

where $\mathrm{O}$ denotes the operator to be preformed, $\set S$ 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 $\Σ[ℕ] n f(n) = 0 + f(0) + f(1) + ⋯$ . 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

$\mathrm ∀$ $\land$ $⊤$ $\mathrm ∃$ $\lor$ $⊥$ $\mathrm ∪$ $\cup$ $\emptyset$ $\mathrm ∩$ $\cap$ $\set U$ $\mathrm Σ$ $+$ $0$ $\mathrm Π$ $\;·\;$ $1$

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

Exceptional binding operators

Operator Interpretation

$\mathrm ∄$ not exists $\mathrm ∃!$ unique $\mathrm ∃?$ an element such that $\mathrm ∃‽$ the unique element such that $\mathrm \vert$ substitute $\mathrm S$ set builder $\mathrm L$ limit $\mathrm ∫$ integral $\mathrm max$ maximum $\mathrm argmax$ argument of the maximum

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

$\L[ℕ]n \frac{1}{n} = 0$

The limit is defined using a generalization of the $(ε, δ)$ definition:

$\L[\set S]x f(x) = l ⇔ \∀[\set U(l)]U \∃[\set S]α \∀[\set S ≥ α]β f(β) ∈ U$

$\L[\set S]x = \Eust l \∀[\set U(l)]ε \∃[\set S]δ \∀[\set S ≥ δ]x ε ∋$

where $\set U(l)$ is the set of all neighborhoods around $l$ and $\set S ≥ α$ is defined as $\S[\set S] β β≥ α$ .

A Riemann integral is defined as

$\∫[{[0,1]}]x f(x) = \L[ℕ]n \frac 1 n \Σ[{[0,n-1]}]i f\g{\frac i n}$

Using the substitute operator this can be written concisely as

$\∫[{[0,1]}]x = \L[ℕ]n \frac 1 n \Σ[{[0,n-1]}]i \|[\frac i n]{x}$

The fundamental theorem of calculus can be written as:

$\∫[{[a,b]}]x = \g{ \|[b]x - \|[a]x } \D[-1]x$

Non binding operators

$\begin{aligned} &\∃[\set S]x P(x) &&⇔& \S[\set S]x P(x) &≠ ∅ \\ &\Eu[\set S]x P(x) &&⇔& \norm{\S[\set S]x P(x)} &= 1 \end{aligned}$

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

Integral transforms

$\e$ $\i$ $\π$ .

$\hat f(ω) = \∫[ℝ]t \e^{-2 \π \i ω t} f(t)$

$\mathcal F_ω^t = \∫[ℝ]t \e^{-2 \π \i ω t}$

$\mathcal L_s^t = \∫[{[0,∞)}]t \e^{-s t}$

$\hat f(ω) = \mathcal F_ω^t f(t)$

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

Generalized:

$\mathcal T(K, \set D)_u^t = \∫[\set D]t K(u, t)$

Extended real numbers

The extended real numbers,

$\overline ℝ = \setc{-∞, +∞} ∪ ℝ$

Set building

Given a proposition $P : \set S → 𝔹$ we can the set builder operator is defined by:

$\set Q = \S[\set S] x P(x) ⇔ \∀[\set S] x x ∈ \set Q ↔ P (x)$

$\∀[ℝ→ℝ] f \∃[ℝ] y \∫[{[0,2,4,…]}] x f(x) = \Σ[ℝ] x f(x)$

$\∫ x f(x) =$

$123 \∂ z 2 \D y \δ x f(x)$

The Laplacian operator:

For some rectalinear basis $\set B$ :

$\begin{aligned} \vec ∇ &= \Σ[\set B]{\vec x} \vec x \∂{\vec x} \\ Δ &= {\vec ∇}^2 = \Σ[\set B]{\vec x} \∂[2]{\vec x} \end{aligned}$