Pure Mathematics

Set Partition

The partition of sets is a very important subject, especially in probability theory, Lesbegue integral, among others. In the previous post we saw equivalence relations and they are directly related to the notion of partition of sets. Definition Definition (Partition) Let $A$ be a nonempty set ($A\neq \emptyset$) and a family $\mathcal{F}$ of nonempty subsets of $A$. The partition definition of $A$ is valid if the following statements are true: (a) any two elements of $\mathcal{F}$ are either equal or disjoint;

November 7, 2022 Read

Equivalence Relations

One of the most important relations in modern mathematics are equivalence relations. By the way, they appear in a large part of mathematics studies. Equivalence Relation In the previous post, we showed some properties such as: reflective; symmetrical; transitive; anti-symmetric. Some of them are necessary to be an equivalence relation. Definition (Equivalence Relations): A relation $\mathcal{R}$ on a set $A$ is said to be an equivalence relation on $A$ if $\mathcal{R}$ is reflexive, symmetric and transitive.

November 6, 2022 Read

Relationship Properties

One of the most important relations in modern mathematics are equivalence relations. For this important properties must be shown. Relationship Properties We start the section with the properties of a relation $\mathcal{R}$ on a set $A$. As a starting point, here is the definition: Definition Let $\mathcal{R}$ be a relation on a set $A.$ Then we say that: $\mathcal{R}$ is reflexive if the condition: $(\forall x \in A)(x\mathcal{R} x)$ is true if for all $x \in A, (x, x) \in R.$ $\mathcal{R}$ is symmetric if the condition: $(\forall x,y \in A)(x\mathcal{R} y \longrightarrow y\mathcal{R} x)$ is true. if for all $x, y \in A,$ if $(x, y) \in \mathcal{R} ,$ then $(y, x) \in \mathcal{R} .$ $\mathcal{R}$ is transitive if the condition: $(\forall x, y, z \in A)(x\mathcal{R} y \wedge y\mathcal{R} z \longrightarrow x\mathcal{R} z)$ is true if for all $x, y, z \in A,$ if $(x, y) \in \mathcal{R} $ and $(y, z) \in \mathcal{R},$ then $(x, z) \in \mathcal{R}.$ $\mathcal{R}$ is anti-symmetric if the condition: $(\forall x, y \in A)(x\mathcal{R} y \wedge y\mathcal{R} x \longrightarrow x = y)$ is true if for all $x, y \in A,$ if $(x, y) \in \mathcal{R}$ and $(y, x) ∈ \mathcal{R},$ then $x = y.$ Examples Below we present some examples of relationship properties:

November 4, 2022 Read

Introduction to Relationships

In the previous post we talked about ordered pairs, Cartesian product. With it, we established the mathematical theory of relationships, using set theory. Motivation Initially, we consider the set $A \times B,$ in which $A = \\{\text{mulheres} \\}$ $B = \\{\text{homens} \\}$. In this case, we can make the sentence: Lucas is Rosana’s husband. Relationship of “being the husband of” Ordered pair $(a,b)$ $a = $ Lucas, $b = $ Rosana. Note that the pair $(b, a)$ does not belong to the relationship, as Rosana is not Lucas’s husband. If the relationship was “being married to”, then both couples would be in the relationship. Formalization Definition (Relation) A relationship between two sets $A$ and $B,$ denoted by $\mathcal{R}(A, B),$ is a subset of $A \times B.$

November 3, 2022 Read

Cartesian Product of Sets

In set theory, one of the subjects we can address is the Cartesian Product of two Sets. In it we talk about ordered pairs, the pair axiom and the Cartesian product. Ordered pair Before we address the subject of the Cartesian product of two sets, we need to talk about ordered pairs. Let $A$ and $B$ be two arbitrary sets. For $a \in A$ and $b \in B,$ and using the specification axiom we can construct the set

November 2, 2022 Read

Operations with Sets

In number theory or arithmetic we can use operations such as addition, subtraction, multiplication and division. In set theory the operations are union, intersection, difference. Some of them defined in the previous post. In this post, we will talk about results for operations between sets. Difference between sets Initially, let’s talk about the difference between sets. To do this, the following definition follows: Definition (Difference between sets): Let A and B be two sets, let us denote the difference between the sets $A$ and $B,$ by $A \setminus B$ or $A−B,$ is the set formed by the elements that are in $A$ and are not in $B.$. Symbolically, we write: $$A - B = \\{x; \quad x \in A \wedge x \notin B\\}.$$

November 1, 2022 Read

Set Family

In set theory, it is important to work with the notion of a family or indexed collection of sets. In the theory of stochastic processes, for example, this concept is widely used, mainly for filtering theory in sigma-algebras. Indexed collection Definition (Indexed family of sets) Let $\Gamma$ be a set, we assume that for each element $\gamma \in \Gamma$ a set $A_{\gamma}.$ is associated. The collection of such sets $A_{\gamma}$ is said to be an indexed family of sets, indexed by the set $\Gamma$ and denoted by $$\\{A_{\gamma}; \gamma \in \Gamma \\}$$

November 1, 2022 Read

Peano axioms

When talking about natural numbers, an axiomatic definition for natural numbers is necessary. To do this, we will show how to define the basic operations for natural numbers. Peano axioms Initially we will axiomatically define the set $\mathbb{N}$ of natural numbers. To describe such properties, we will use the symbols 0 (zero) and s (successor). (a) $0 \in \mathbb{N};$ Zero belongs to the set of natural numbers. (b) $\forall n \in \mathbb{N}\quad s(n) \in \mathbb{N};$ For every number belonging to the set of naturals, the successor belongs to the natural ones. (c) $\forall n \in \mathbb{N}\quad s(n) \neq 0;$ For every number belonging to the set of natural numbers, the successor is different from zero. (d) $\forall m,n \in \mathbb{N}\quad s(n)=s(m) \Rightarrow n=m;$ For every two pairs of numbers belonging to the natural set, such that the successors are equal, this implies that these numbers are equal. (e) Let $P(x)$ be a property on elements of $\mathbb{N}.$ Apply: (i) $P(0);$ (ii) $\forall n \in \mathbb{N}, \quad P(n)\Rightarrow P(s(n))$ Then it is valid $P(n)$ for all $n \in \mathbb{N}.$ The axiom $(e)$ is known as the principle of induction. The set $\\{0, 1, 2, \dots\\}$ satisfies the above properties if we interpret the symbol $s(n)$ as $n + 1.$ To note that the principle of induction holds in such a set, suppose that it does not hold. Then there is $n$ the smallest such that is not worth P(n).

October 30, 2022 Read

Introduction to Set Theory

In set theory there are two wings of study, the axiomatic and the naive theory. The axiomatic wing relies on an informal understanding of sets as collections of objects. The naïve approach, well developed at the end of the 19th century, treats sets as demonstrable forms with propositions, lemmas, theorems and corollaries. Set theory is very important in mathematics, especially for modern theory such as: numbers relations functions Specification axiom Axioms are fundamental to mathematics. However, when we assume that any operations on sets can lead to paradoxes such as Russell’s paradox. Axiomatic theory was fundamental in determining which operations are allowed for sets.

September 3, 2022 Read

Primitive Notions

In this post we will study the primitive elements of mathematics, which are fundamental to the elements of mathematics. This chapter is dedicated to set theory, however, let’s take it slowly. In each post we will talk a little about this complex and dense theory. In set theory it is necessary to have some primitive notions: primitive notions of set. primitive notions of membership, which we use the symbol belongs $\in$. Relevance In relevance, the relationship between sets can be established as follows:

August 20, 2022 Read

Demo

In previous posts we defined porpositions, theorems, corollaries, among others. To demonstrate theorems, it means that the given proposition is a tautology, in which we obtain through the truth tables. Below, we present some methods for demonstrating results. Deductive method The deductive method is obtained through the use of definitions and other pre-established results. For this, we use the implications $(\Rightarrow)$ and logical bi-implications $(\Leftrightarrow)$. Example 1: With the following statements, we will show whether it is a theorem or not. For this, the following hypotheses are: $H_1$: time is money. $H_2$: bum has a lot of time. And the thesis: $T$: Tramp has a lot of money. Is the proposition $H_1 \wedge H_2 \Rightarrow T$ a theorem?

August 14, 2022 Read

Theorems

In previous posts we talked about postulates and axioms, without a formal demonstration of results. Here we begin the concepts of formal demonstration, for this, we present some fundamental concepts to begin the topic. Concepts The first very important concept in mathematics is definition. Definition: is a statement that describes the meaning of a mathematical term. Proposition In previous posts we defined this concept, but now, we present it in a simpler way:

August 14, 2022 Read