Symmetric Relation
This definition is more clearly described sybollically: ∀a,b∈X(aRb⟺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
Set-Theory
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. ∀x,y,z∈X,ifxRyandyRz,thenxRz
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:
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Reflexive Relation
A binary relation R over a set X is reflexive if it relates every x∈X to itself. ∀x∈X|xRx
An example of a reflexive relation is “is equal to” since any number within R would be mapped back to itself.
Tao’s definition:
Given any object x, we have x=x.
See also:
Symmetric Relation
Transitive Relation
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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∈Y, there exists x∈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×B. …alternatively… Let A and B be two non-empty sets, then every subset of A×B defines a relation from A to B and ever relation from A to B is a subset of A×B. Let R⊆A×B and (a,b)∈R. ...
Neighborhood
The set of points surrounding a point p in a set V, such that p∈Rn, within a radius ϵ>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≠x′⟹f(x)≠f(x′) i.e. Given a set X and a set Y, no two elements (say, x and x′) 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⟶Y from X to Y.