Glossary

This page will list all terms introduced in the other sections. Concepts that can't be explained in a few lines will instead be links.

Maths

Associativity

For an operation to be associative, it means that you can change the order you do each operation around while keeping the same answer.

Simple addition is associative. Adding $a$ and $b$ together before adding the sum to $c$ results in the same answer as adding $b$ and $c$ together and then adding that sum to $a$.

Example

$$\begin{aligned} (4 + 2) + 5= 6 + 5 &= 11 \\ &= 4 + 7 = 4 + (2 + 5) \end{aligned}$$

General Case

$$(a+b)+c = a + b + c = a + (b + c)$$

Commutativity

For an operation to be commutative, it means that the order of the elements does not matter.

Simple addition is commutative. Adding $a$ to $b$ is the same as adding $b$ to $a$.

Example

$$6+4 = 10 = 4 + 6$$

General Case

$$a + b = b + a$$

Simple subtraction is not commutative: subracting $a$ from $b$ is not the same as subracting $b$ from $a$.

Example

$$\begin{gather*} \begin{aligned} 6-4 = 6 + (-4) &= 2 \\ 4-6 = (-6) + 4 &= -2 \end{aligned} \\\\ -2 \neq 2 \end{gather*}$$

General Case

$$\begin{gather*} \begin{aligned} a-b &= a + (-b) \\ b-a = (-a) + b &= -a + b \end{aligned} \\\\ a + (-b) \neq -a + b \end{gather*}$$

Distributivity

Distributivity describes how one operation acts over another operation. There can be both left and right distributivity, and the following must be true for them to hold:

Left

$$a\star (b\oplus c) = (a\star b) \oplus (a\star c)$$

Right

$$(b\oplus c) \star a = (b\star a) \oplus (c\star a)$$

Identity

An identity element is an element that does nothing when operated on with another element. An operation can have both left and right Identities.

Inverse

An inverse element is an element, that when operated on with another element, will result in the identity element of that operation. There can be both left and right inverses.

Left

$$e^{-1}_l \star a = e^{id}_l$$

Right

$$a \star e^{-1}_r = e^{id}_r$$