$\mathbb{N}$ is the set of Natural numbers (positive integers including zero).

$\mathbb{Z}$ is the set of Integers numbers (also negative).

$\mathbb{Q}$ is the set of Rational numbers (fractionals).

$\cup$ is the union operator symbol.

$\cap$ is the intersection operator symbol.

$\setminus$ is the union difference symbol.

$\in$ is the "belongs to set" symbol.

$\subset$ is the "strict subset of" symbol. (inverted in $\supset$)

$\subseteq$ is the "subset of or equal" symbol. (inverted in $\supseteq$)

$|A|$ is the "cardinality" of set A, how many elements A is composed of.

$|A\times B|$ is the "cartesian product" of sets A and B. It is made of elements $(a,b)$, where $a \in A$ and $b \in B$, with all possible combinations.

$a R b$ is equal to $(a,b) \in R$, and R is a relation, a subset of the cartesian products of the sets where a and b take their values from. It can also be expressed as $R(a,b)$.

A statement "A" can be only True or False

$\lor$ is logical or

$\land$ is the logical and

$\lnot A$ indicates the negation of "A"

$\exists x(statment)$ indicates that the "x" present in the statement is a variable that makes the statement True for "at least" a value of x.

$\forall x(statment)$ indicates that the "x" present in the statement is a variable that makes the statement True for "all possible" values of x.

$A \iff B$ is a "if and only if", making A and B having the same truth value.

$A \rightarrow B$ is a "A implies B", when A is True, B has to be True as well.

Commutativity: $a \circ b = b \circ a$

Associativity: $x \circ (y \circ z) = (x \circ y) \circ z = x \circ y \circ z$

Identity element e: $\exists e \forall x (x \circ e = e \circ x = x)$

Zero element e: $\exists e \forall x (x \circ e = e \circ x = e)$

Distributive: $x \circ (y \Box z) = (x \circ y) \Box (x \circ z)$

Inverse: $x \circ (y \Box z) = (x \circ y) \Box (x \circ z)$

Addition: Commutative, Associative, Identity on $0$

Subtraction: Inverse of Addition

Multiplication: Commutatiive, Associative, Identity on $1$, Zero element on $0$, Distributive over Addition

Division: Inverse of Multiplication, Distributive over Addition (and Subtraction)