Notation

Remco Bloemen

2015-08-26

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Building sets

𝒮×𝒮=𝒮2 \mathcal S \times \mathcal S = \mathcal S^2

Notation

Binding operators

Binding operators have the generic notation

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

where O\mathrm{O} denotes the operator to be preformed, 𝒮\mathcal S is a set from which elements xx will be taken. The operator is right-associative and hence applied on the expression following it, which may include references to xx. The resulting expression will not depend on xx, 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 Σnf(n)=0+f(0)+f(1)+\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 𝒰\mathcal U
Σ\mathrm \Sigma ++ 00
Π\mathrm \Pi \;\cdot \; 11

The expression is a function with 𝒮\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
S\mathrm S set builder
L\mathrm L limit
\mathrm \int integral
max\mathrm max maximum
argmax\mathrm argmax argument of the maximum

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

$$ \L[ℕ]n \frac{1}{n} = 0 $$

The limit is defined using a generalization of the (ϵ,δ)(\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 𝒰(l)\mathcal U(l) is the set of all neighborhoods around ll and 𝒮α\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:

[a,b]x=(|bx|ax)Dx1 \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(𝒮)r(\mathcal S) arbitrary element from 𝒮\mathcal S.

Integral transforms

e\mathrm e i\mathrm i π\mathrm \pi .

f̂(ω)=te2πiωtf(t) \hat f(\omega ) = \mathrm{\int }^{t}_{ℝ} \mathrm e^{-2 \mathrm \pi \mathrm i\omega t} f(t)

ωt=te2πiωt \mathcal F_\omega ^t = \mathrm{\int }^{t}_{ℝ} \mathrm e^{-2 \mathrm \pi \mathrm i\omega t}

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

f̂(ω)=ωtf(t) \hat f(\omega ) = \mathcal F_\omega ^t f(t)

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

Generalized:

𝒯(K,𝒟)ut=𝒟tK(u,t) \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:𝒮𝔹P : \mathcal S \rightarrow \mathbb{B} we can the set builder operator is defined by:

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

fy[0,2,4,]xf(x)=Σxf(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)

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

123z2Dyδxf(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:

=ΣxxxΔ=2=Σxx2 \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}