• $\mathbb{N}$ 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).

• $\mathbb{I}$ is the set of Imaginary numbers ($a*i$)

• $\mathbb{R}$ is the set of Real numbers (irrationals, $\sqrt{2},\ldots$)

• $\mathbb{C}$ is the set of Complex numbers ($a+b*i$)

• $\cup$ 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.

• $=$ is the equality symbol.

• $<$ (symmetric of $>$) is the "less than" symbol.

• $\leq$ (symmetric of $\geq$) is the "less than or equal" symbol.

• 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.

• $A \implies B$ 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)

• $a R b$ 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 $R(a,b)$.

• $Eq(A)$ | $=(A)$ | $<(A)$... are implicitly a relation of type $R(A,A)$, as both elements are from the same set.

Property of relations:

• Reflexivity: $\forall x \in A, x \triangle x$

• Irreflexivity: $\forall x \in A, \lnot x\triangle x$

• Symmetry: $\forall x,y \in A, x\triangle y \implies y\triangle x$

• Asymmetry: $\forall x,y \in A, x \triangle y \implies \lnot (y \triangle x)$

• Antisymmetry: $\forall x,y \in A, (x \triangle y \land y \triangle x) \implies (x = y)$

• Transitive: $\forall x,y,z \in A, (x \triangle y \land y \triangle z) \implies (x \triangle z)$

• Connected: $\forall x,y \in A, (x \neq y) \implies (x \triangle y \lor y \triangle x)$

• Strongly Connected: $\forall x,y \in A, (x \triangle y \lor y \triangle x)$

Relation of Equivalence has: "Reflexivity, Symmetry, Transitivity".

Partial Order relation:

• If weak: "Reflexivity, Antisymmetry, Transitivity"

• If strong: "Irreflexivity, Asymmetry, Transitivity"

• $i^2=-1$

• $i^3=-i$

• $i^4=1$

• Addition: $(a+bi) + (c+di) = (a+c) + (b+d)i$

• Multiplication: $(a+bi)*(c+di)=(ac-bd) + (ad+bc)i$