Set-Theory
Symmetric Relation
This definition is more clearly described sybollically: \begin{equation} \forall a,b \in X(aRb\,\iff\,bRa) \end{equation} Tao’s definition: Given any two …
Read →Transitive Relation
A relation \(R\) over a set \(X\) is transitive if for all elements x, y, z in \(X\), whenever \(R\) relates \(x\) to \(y\) and \(y\) to \(z\), then \(R\) also …
Read →Reflexive Relation
A binary relation \(R\) over a set \(X\) is reflexive if it relates every \(x \in X\) to itself. \begin{equation} \forall x \in X \,|\, xRx \end{equation} An …
Read →Math
Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts Bijective Function Injective Function Surjective Function Transitive …
Read →Surjective Function
A function is surjective or “onto” if For every \(y \in Y\), there exists \(x \in X\) such that \(f(x) = y\).
Read →Relations (Sets)
Definition A relation \(R\) from the elements of set \(A\) to the elements of set \(B\) is a subset of \(A \times B\). …alternatively… Let \(A\) …
Read →Neighborhood
The set of points surrounding a point \(p\) in a set \(V\), such that \(p \in \mathbb{R}^{n}\), within a radius \(\epsilon > 0\) See also: Open Balls
Read →Locality (Math)
Locality refers to: A property \(P\), of a point \(x\), which holds true near every point around \(x\). As an example: A sphere (and, more generally, a …
Read →Injective Function
A function is injective (“one-to-one”) if \(x \neq x^{\prime} \Longrightarrow f(x) \neq f(x^{\prime})\) i.e. Given a set \(X\) and a set \(Y\), no …
Read →Cardinality
Equality in Cardinality Two sets \(X\) and \(Y\) are equal iff there exists a bijection \(f : X \longrightarrow Y\) from \(X\) to \(Y\).
Read →Binary Relation
A binary relation \(R\) over sets \(X\) and \(Y\) is a subset of the cartesian product of \(X \times Y\). Where \(X\) is the domain, or set of departure of …
Read →Bijective Function
A function is bijective or “invertible” if it is Both one-to-one and onto (injective and surjective) In other words: each element of one set is …
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