Editing Maths (section)

==Properties==

* '''associative''' - 𝒶(𝒷𝒸) = (𝒶𝒷)𝒸. '''∀𝓍,𝓎,𝓏. 𝓍∘(𝓎∘𝓏)=(𝓍∘𝓎)∘𝓏'''
* '''commutative''' - 𝒶𝒷 = 𝒶𝒷.  '''∀𝓍,𝓎. 𝓍∘𝓎=𝓎∘𝓍'''
* '''Identity element''': There exists an element ℯ such that for each element 𝓍, ℯ∘𝓍 = 𝓍 = 𝓍∘ℯ ;   formally: '''∃ℯ ∀𝓍. ℯ∘𝓍=𝓍=𝓍∘ℯ'''. The identity element of multiplication is 1 as 1×𝓍=𝓍=𝓍×1. The identity of addition is 0 as 0+𝓍=𝓍=𝓍+0.
* '''Inverse element''': It can easily be seen that the identity element is unique. If this unique identity element is denoted by ℯ then for each 𝓍, there exists an element 𝒾 such that 𝓍∘𝒾=ℯ=𝒾∘𝓍;   formally: '''∀𝓍 ∃𝒾. 𝓍∘𝒾=ℯ=𝒾∘𝓍'''. The multiplicative inverse: '''𝒶𝒶⁻¹=1=𝒶⁻¹𝒶'''


* '''domain''' - The set of elements on which a function has a valid definition is it's domain. For '''𝒻: X→Y''', X is the domain.
* '''codomain / target''' - The possible set that a function outputs. For '''𝒻: X→Y''', X is the target. There may be things in the target that aren't actually reachable by the function.
* '''image / range''' - The set of elements that a function maps to and no more. This is like the '''target''' but only the possible elements are in the set. For example the '''cos''' function could be defined as having a target of the real numbers, but the range could only be between -1 and 1. It is not always possible to define the range. Concepts of Modern Mathematics (pg. ~67). Some ambiguity as sometimes 'range' refers to the codomain.


* '''homomorphism''' - A category of function. Preserves the structure. '''𝒻: X→Y'''. '''𝒻(a)𝒻(b)=𝒻(c)''' and '''𝒶𝒷=𝒸'''. A homomorphism of vector spaces is referred to as a '''linear map'''.
* '''injection''' - Each target element must be reachable from only one element in the domain. It does not need to be '''surjective''', meaning that it's possible to have elements in the target that aren't reachable at all. But you can not have an element in the domain that is reachable from more than one target element. cos is not an injection because an infinite number of inputs get mapped to values between -1 and 1.
* '''surjection (onto)''' - If every element of a target set T is reachable by a function, that function is a surjection '''onto''' T. This means the target is also the range. Does not need to be an injection. - Concepts of Modern Mathematics (pg. 70).
* '''bijection''' - A bijection is invertible. It is an '''injection''' and a '''surjection'''. A function which relates each member of a set S (the domain) to a separate and distinct member of another set T (the range), where each member in T also has a corresponding member in S. A mapping that is both one-to-one (an injection) and onto (a surjection).
* '''isomorphism'''  - A '''homomorphism''' that is also a '''bijection'''. An operation-preserving bijection. A homomorphism with an inverse that is also a homomorphism. '''𝒻: X→Y, ℊ: Y→X'''
* '''automorphism''' - Galios Groups related. An isomorphism from a structure to itself.
* '''Abelion''' - An abelian group/operation is '''commutative'''.