To qualify as a vector space, a set V and its associated operations of addition (+) and multiplication/scaling (⋅) must adhere to the below: Associativity # u+(v+w)=(u+v)+w
An property (of a mathematical object) is invariant if, after some operation(s) are applied, that property remains unchanged. For instance, in a geometrical space where the concept of “length” is defined (by some metric); a physical object, say a pencil, will maintain its characteristics (length) despite a change of coordinates (e.g. polar to cartesian). In short, vectors are invariant, but their components are not (under a transformation).
A manifold is a topological space that locally resembles Euclidean space near each point. As defined in Physics # An N-dimensional manifold of points is one for which N independent real coordinates (x1,x2,…,xN) are required to specify a point completely. These N coordinates are denoted collectively by xa, where it is understood that a=1,2,…,N. As an example, in R2 we have a 2-dimensional manifold of points descrived by the real coordinates (x1,x2). ...
This definition is more clearly described sybollically: ∀a,b∈X(aRb⟺bRa)
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. ∀x,y,z∈X,ifxRyandyRz,thenxRz
A binary relation R over a set X is reflexive if it relates every x∈X to itself. ∀x∈X|xRx
Concepts # Relations Cardinality Binary Relation Symmetric Relation Transitive Relation Cartesian Product Reflexive Relation Surjective Function
Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts # Bijective Function # Injective Function # Surjective Function # Transitive Relation # Symmetric Relation # Locality # Peano Axioms #
For any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. z≤x+y
Concepts # Metric Space Open Balls Neighborhood Manifold Topological Space