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 objects \(x\) and \(y\) of the same type, if \(x = y\), then \(y = x\). See also: Transitive Relation Reflexive 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 relates \(x\) to \(z\). \begin{equation} \forall x,y,z \in X,if\,xRy\,and\,yRz,\,then\,xRz \end{equation} Tao’s definition: Given any three objects x, y, and z of the same type, if \(x = y\) and \(y = z\), then \(x = z\). See also: ...
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 example of a reflexive relation is “is equal to” since any number within \(\mathbb{R}\) would be mapped back to itself. Tao’s definition: Given any object x, we have \(x = x\). See also: Symmetric Relation Transitive Relation ...
Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts # Bijective Function # Injective Function # Surjective Function # Transitive Relation # Symmetric Relation # Locality # Peano Axioms #
A function is surjective or “onto” if For every \(y \in Y\), there exists \(x \in X\) such that \(f(x) = y\).
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 \(A\) and \(B\) be two non-empty sets, then every subset of \(A \times B\) defines a relation from \(A\) to \(B\) and ever relation from \(A\) to \(B\) is a subset of \(A \times B\). Let \(R \subseteq A \times B\) and \((a, b) \in R\). ...
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
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 manifold) is locally Euclidean. For every point on the sphere there is a neighborhood which is identical to Euclidean space.
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 two elements (say, \(x\) and \(x^{\prime}\)) from \(X\) map to the same element in \(Y\). (Note: a function can be both injective and surjective, this image is injective only)
Equality in Cardinality # Two sets \(X\) and \(Y\) are equal iff there exists a bijection \(f : X \longrightarrow Y\) from \(X\) to \(Y\).