# Overview

The follow is a list of mathematical tools. They’re in alphabetical order and contain reference links. I do not provide examples in this list; I’ve decided to only give definitions in order to minimize the reference list, leaving interesting example for blog posts.

## <a name="abelian> Abelian group

When the operation for a group commutes, we say that the group is *Abelian*.

## Closure

We say a set has *closure* for an operation operation when the operation always results in giving a member of the set.

## Commutative

We call an operation *commutative* when the left and right hand sides swap without changing the result: $$2 \times 3 = 3 \times 2$$

## Group

A *group* gives mathematical objects well-defined properties with respect to an operation $\cdot$. These properties are:

- closure: for any two objects $a$ and $b$ in the group, there exists some object $c$ such that $a \cdot b = c$
- associativity: for any three objects $a$, $b$, $c$ in the group, the operation gives the same result when applied left to right $( a \cdot b ) \cdot c$ or right to left $a \cdot ( b \cdot c )$
- identity: there exists some object $e$ in the group that, for any object $a$, $a$ is the result of the operation $e \cdot a = a$
- inverse: for any object $a$ in the group, there exists an object $x$ where the identity $e$ is the result of the group operation $a \cdot x = e$

## Identity

Identity may refer to either a mathematical object or an operation.

Identity as an operation can be described as an operation that does nothing when applied to an object.

When referring to a mathematical object, the precise definition further depends on its sister objects and an operation. The idenity $e$ simply satisfies the rule $a \cdot e = a$ – any particular representation of $e$ depends on what $a$ looks like and what $\cdot$ does…

## Operation

When we do something with a mathematical object, we call that performing an *operation*. Some common operations are:

- $+$ addition
- $-$ subtraction
- $\times$ multiplication
- $\div$ division

a few advanced but still common operations are

- $\otimes$ tensor
- $\cdot$ used as an alternative for multiplication, or for the vector dot product
- $\langle,,\rangle$ inner product

Keep in mind that the mathematical symbol may have a different definition for its corresponding operation depending on the context – in some contexts, the operation may not be defined at all! For example $\div$ makes sense with real numbers, complex numbers, and rational numbers, but $\div$ does not always make sense with natural numbers, integers, or vectors and matrices.