Zettelkastens

Open Set

The elements of a topological space are called open sets.

Homomorphism

Given two magmas $A$ and $B$, a function $\phi : A \to B$ is a homomorphism if and only if $\phi(x \cdot y) = \phi(x) \cdot \phi(y)$, for all $x, y \in A$.

Intersection

The intersection between two sets $A$ and $B$ is a set that contains all the elements that are in $A$ and in $B$. The intersection of sets $A$ and $B$ is denoted as $A \cap B = \{x | x \in A \text{ and } x \in B\}$.

Union

The union between two sets $A$ and $B$ is a set that contains all the elements that are in $A$ or in $B$. The union of sets $A$ and $B$ is denoted as $A \cup B = \{x | x \in A \text{ or } x \in B\}$.

Topological Space

A topological space or topology on a set $X$ is a collection $\tau$ of subsets of $X$ which is closed under:

• finite intersections
• arbitrary unions

Endomorphism

An endomorphism is a morphism whose domain and codomain are equal.

Ring Homomorphism

A ring homomorphism is a function $f : R \to S$ from a ring $R$ to a ring $S$ that has the following properties, for all $a, b \in R$:

Endomorphism Ring

Given an abelian group $A$, the ring produced by the set of endomorphisms of $A$ with addition of morphisms defined as $f + g \mapsto (x \mapsto f(x) + g(x))$ and multiplication of morphisms defined as $f \cdot g \mapsto (x \mapsto f(g(x)))$ is called the endomorphism ring of $A$. Additionally, it is denoted with $\text{End}A$.

Cone

Given a point and a space curve, a cone is the union of infinite lines passing through that point and curve.

Conic Section

Given cone and a plane, a conic section is the intersection between that cone’s surface and that plane.

Locus

A locus is a set of points that satisfy a given predicate.

Point

A point is a position in abstract space.

Hyperbola

Given points $F_1$, $F_2$ and a distance $2a$, a hyperbola is the locus of the predicate $P(p): \lvert \text{dist}(p, F_1) - \text{dist}(p, F_2) \rvert = 2a$.

Solution Set

The solution set for an equation on $n$ variables is the set of $n$-tuples where each entry in the $n$-tuple corresponds with a variable, and for each tuple, substituting each variable with its corresponding entry in the tuple will make the equation true.

Subset

A subset of a set $X$, is a set $Y$ where each element of $Y$ is in $X$. In other words,

Cartesian Plane

The Cartesian plane is the set obtained from taking the cartesian product of $\mathbb{R}$ with itself.

Remarks

Conveniently, each tuple in this set can be represented on a 2-dimensional plane where the horizontal axis consists of the first set of real numbers and the second axis consists of the second set of real numbers.

Parabola

Given line $l$ and a point $F$, a parabola is the locus of the predicate $P(p): \text{dist}(p, F) = \text{dist}(p, l).$

Ellipse

Given points $F_1$ and $F_2$, called the foci (singular: focus), and a distance $2a$ where $\text{dist}(F_1, F_2) < 2a$, an ellipse is the locus for the predicate $P(p): \text{dist}(p, F_1) + \text{dist}(p, F_2) = 2a$.

Circle

A circle centered at point $(a, b)$ with radius $r$ is the locus that satisfies the equation $(x - a)^2 + (y - b)^2 = r^2$.

Absolute Value

For real numbers, the absolute value of a number is simply the distance from $0$. However, absolute value can be generalized for rings, 4b4b4a1b and vector spaces.

Cartesian Product

The Cartesian product of two sets $A$ and $B$, is the set obtained by creating tuples $(a, b)$ for all $a \in A$ and for all $b \in B$. In other words,

Comparability

For a given partial order $R$; whenever $(a, b) \in R$, then the elements $a$ and $b$ are said to be comparable.

Antisymmetry

A homogeneous binary relation $R$ is antisymmetric if and only if $(a, b) \in R$ and $a \ne b \implies (b, a) \notin R$, for all $(a, b) \in R.$

Total Order

A total order is a partial order in which every two elements are comparable.

Partial Order

A partial order is a homogeneous binary relation that is:

• reflexive
• antisymmetric
• transitive

Tranisitivity

A homogeneous binary relation $R$ is transitive if and only if $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$, for all $(a, b), (b, c) \in R.$

Inequality

Given a partial order $R$, whenever two elements are comparable, they are said to be non-equal.

Homogeneous Binary Relation

A homogeneous binary relation is a binary relation from a set to itself.

Symmetry

A homogeneous binary relation is symmetric if and only if $(a, b) \in R \implies (b, a) \in R$, for all $(a, b) \in R$.

Reflexivity

A homogeneous binary relation $R$ over a set $A$ is reflexive if and only if $\forall a \in A, (a, a) \in R.$

Binary Relation

A binary relation $R$ is a relation over two sets $A$ and $B$. Then, $R \subseteq \{(a, b) | a \in A, b \in B\}$.

Predicate

A predicate is a statement whose veracity depends on a variable.

Remarks

A predicate is usually written as a function; taking as a parameter the mathematical object necessary to determine the veracity of the statement.

Veracity

Veracity is the property of “being in accordance with reality” or “not being in accordance with reality”. We call “being in accordance with reality”, “true” and otherwise “false”.

Statement

A statement is, most times, defined to be a sentence that asserts a property. In order to be a well-formed statement, one must be able to logically conclude its veracity.

Real Numbers

#todo

Irrational Numbers

An irrational number is a number with a decimal expansion that cannot be expressed as the [[2b1c]] of two [[10a]] #todo

Rational Numbers

A rational number is a number that can be expressed as the division of two integers the second integer being nonzero. It is denoted as the set $\mathbb{Q}$ and can be expressed as $\mathbb{Q} = \{\frac{a}{b} | a, b \in \mathbb{Z}, b \ne 0\}.$

Integers

An integer is a natural number that is either positive or negative. It is denoted as the set $\mathbb{Z}$, and can be expressed as: $\mathbb{Z} = \{-x | x \in \mathbb{N}\} \cup \mathbb{N}_0.$

Linear Equation

A linear equation is an equation of several variables where each coefficient does not contain any of the variables.

Variable

A variable, usually denoted with a single letter, expresses an unspecified mathematical object.

Coefficient

A coefficient is a multiplicative factor in some expression.

Equation

An equation is expresses that two mathematical objects are equal. This will be denoted with the symbol $=$.

Natural Numbers

A natural number is a number used for counting things in everyday life. It is denoted as the set $\mathbb{N}$ and can be expressed as: $\mathbb{N} = \{1, 2, \dots\}.$

Absorbing Element

Let $(S, \cdot)$ be a magma. A zero element is an element $z \in S$ such that $s \cdot z = z \cdot s = z, \forall s \in S.$ $z$ is considered a left-zero when $z \cdot s = z$ and a right-zero when $s \cdot z = z$.

Rng

A rng is a set $R$ equipped with two binary operations $+ : R \times R \to R$ and $\cdot : R \times \to R$ such that:

Zero Ring

A zero ring is a ring where its underlying set is the single element set $\{0\}$ with the binary operations $+$ and $\cdot$.

Semiring

A semiring is a set $R$ equipped with two binary operations $+ : R \times R \to R$ and $\cdot : R \times \to R$ such that:

Algebraic Structure

An algebraic structure consists of:

• a nonempty set $A$
• a collection of operations on $A$ that satisfy a finite set of axioms.

Group-like Structures

Set

A set is a collection of things.

Function

A function from a set $X$ to a set $Y$ assigns to each element in $X$ a single element in $Y$.

Totality

A function from set $X$ to set $Y$, noted as $f : X \to Y$, is total if the domain for which $f$ is defined is equal to $X$. On the contrary, if $f$ is not defined for some element in $X$, then $f$ is said to be a partial function.

Operator

An operator is a function from a set to itself.

Binary Operator

A binary operator is an operator of arity two.

Associativity

A binary operator, $\cdot: A \times A \to A$ is associative if and only if $(a \cdot b) \cdot c = a \cdot (b \cdot c), \forall a, b, c, \in A.$

Identity

An identity element in a set $X$ with a binary operator $\cdot: X \times X \to X$ is an element $e \in X$ such that $e \cdot x = x \cdot e = x, \forall x \in X.$

Divisibility

A set $X$ with a binary operator $\cdot: X \times X \to X$ has the divisibility property if $\forall a, b \in X, \exists x, y \in X : a \cdot x = b$ and $y \cdot a = b.$

Commutativity

A set $X$ with a binary operator $\cdot: X \times X \to X$ is commutative if $\forall a, b \in X, a \cdot b = b \cdot a.$

Closure

Let $f$ be an operator that produces some elements in a set $S$ by combining 1 or more elements of $S$. A subset $X$ of $S$ is said to be closed if the set of elements produced by $f$ by operating it on all elements of $X$, is equal to $X$.

Left Distributivity

Given a Set $S$ and two binary operators $*$ and $+$ on $S$. If $\forall x, y, z \in S, x * (y + z) = (x * y) + (x * z)$, then $*$ is left-distributive.

Distributivity

Given a set $S$ and two binary operators $*$ and $+$ on $S$. If $*$ is left distributive and right distributive, then $*$ is said to be distributive.

Right Distributivity

Given a Set $S$ and two binary operators $*$ and $+$ on $S$. If $\forall x, y, z \in S, (y + z) * x = (y * x) + (z * x)$, then $*$ is right-distributive.

Arity

We define arity to be the number of arguments, operands or parameters taken by a function, operation or relation.

Tuple

A tuple is a set of mathematical objects in which the order of the elements is significant.

Relation

Given sets $X_1, X_2, \dots, X_n$, a relation $R$ over $\Pi_{i = 1}^{n}X_i$ is a set of $n$-tuples where the $i$th element is an element of $X_i$. In other words,

Partial Magma

A set $X$ with a partial binary operator $\dot: X \times X \to X$ is a partial magma.

Semigroupoid

Let $X$ be a set. Define $\forall a, b \in X, \text{Mor}(a, b)$ the set of morphisms from $a$ to $b$. Additionally, if $f \in \text{Mor}(a, b)$ then we write, $f : a \to b$.

Category

A category $C$ consists of:

• a class of objects denoted as $\text{ob}(C).$
• a class of morphisms denoted as $\text{mor}(C).$
• a domain class function $\text{dom}: \text{mor}(C) \to \text{ob}(C).$
• a codomain class function $\text{codom} : \text{mor}(C) \to \text{ob}(C).$
• for every three objects $a, b, c \in \text{ob}(C)$, a binary operator $\circ: \text{hom}(a, b) \times \text{hom}(b, c) \to \text{hom}(a, c).$ Let $a, b, c, d \in \text{ob}(C),$ and $f \in \text{hom}(a, b), g \in \text{hom}(b, c), h \in \text{hom}(c, d)$, the following properties must hold:
• $h \circ (g \circ f) = (h \circ g) \circ f.$
• $\exists 1_a \in \text{hom}(a, a) : 1_a (a) = a.$
• $1_b \circ f = f \circ 1_a = f.$

Remarks

A category is also a semigroupoid with identities for every element.

Groupoid

A groupoid is a nonempty set $G$ with a unary operation ${}^{-1}: G \to G$, and a partial function $* : G \times G \to G$. Let $a, b, c \in G$, the following properties must hold:

Commutative Groupoid

A commutative groupoid is a groupoid where the binary operator is commutative.

Magma

A Partial Magma where the binary operator is total is a magma.

Commutative Magma

A commutative magma is a magma where the binary operator is commutative.

Quasigroup

A quasigroup is a set with a binary operator where the operator has the divisibility property.

Commutative Quasigroup

A commutative quasigroup is a quasigroup in which the binary operator is also commutative.

Loop

A loop is usually defined as a quasigroup with an identity element.

Remarks

A loop can also be defined as a unital magma with divisibility.

Loop

A commutative loop is a loop that is also commutative.

Associative Quasigroup

An associative quasigroup is usually defined as a quasigroup that is also associative.

Remarks

An associative quasigroup is also a semigroup with inverses.

Commutative and Associative Quasigroup

A commutative-and-associative quasigroup is a quasigroup that is both commutative and associative.

Unital Magma

A unital magma is a magma that has an identity element in its underlying set.

Commutative Unital Magma

A commutative unital magma is a unital magma that is also commutative.

Semigroup

A semigroup is an associative magma.

Commutative Semigroup

A commutative semigroup is a semigroup that is also commutative.

Monoid

A monoid is a semigroup with an identity.

Remarks

In category theory, a monoid is a category with 1 object.

Commutative Monoid

A commutative monoid is a monoid that is also commutative.

Remarks

This is also called, less commonly, an abelian monoid.

Group

A group is a non-empty set $G$ together with a binary operation $\cdot: G \to G$. The following three requirements called the group axioms have to be met:

Abelian Group

An abelian group is a group that is also commutative.

Ring

A ring is a semiring where each element has an additive inverse.

Remarks

A ring is a set $R$ equipped with two binary operations $+$ and $\cdot$ satisfying the following three sets of axioms, called the ring axioms:

Commutative Ring

A commutative ring is a ring in which the multiplication operation is commutative.

Integral Domain

An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

Morphism

A morphism is a mapping between one mathematical object and another mathematical object.

Class

A class is a collection of sets. When a class is not a member of any other mathematical object, it is called a proper class.