Set-Theory

Symmetric Relation

This definition is more clearly described sybollically: $$\forall a,b \in X(aRb\,\iff\,bRa)$$ 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

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 relates $$x$$ to $$z$$. $$\forall x,y,z \in X,if\,xRy\,and\,yRz,\,then\,xRz$$ 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: ...

Reflexive Relation

A binary relation $$R$$ over a set $$X$$ is reflexive if it relates every $$x \in X$$ to itself. $$\forall x \in X \,|\, xRx$$ 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 ...

Math

Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts # Bijective Function # Injective Function # Surjective Function # Transitive Relation # Symmetric Relation # Locality # Peano Axioms #

Surjective Function

A function is surjective or “onto” if For every $$y \in Y$$, there exists $$x \in X$$ such that $$f(x) = y$$.

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 $$A$$ 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 $$(a, b) \in R$$. ...

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

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 manifold) is locally Euclidean. For every point on the sphere there is a neighborhood which is identical to Euclidean space.

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 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)

Cardinality

Equality in Cardinality # Two sets $$X$$ and $$Y$$ are equal iff there exists a bijection $$f : X \longrightarrow Y$$ from $$X$$ to $$Y$$.