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 \(Y\).