% Remco Bloemen % 2015-08-26

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Building sets

$$ \set S × \set S = \set S^2 $$


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

$\mathrm ∀$ $∧$ $⊤$ $\mathrm ∃$ $∨$ $⊥$ $\mathrm ∪$ $∪$ $∅$ $\mathrm ∩$ $∩$ $\set U$ $\mathrm Σ$ $+$ $0$ $\mathrm Π$ $\;·\;$ $1$

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

$\e$ $\i$ $\π$.

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

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

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

$$ \hat f(ω) = \mathcal F_ω^t f(t) $$


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

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 $\set B$:

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

Remco Bloemen
Math & Engineering