At this point in the posts, we will talk about sets as a collection of elements, which can be abstract or not. The abstract ones are those that are in the mathematical context, such as a set of odd integers, among others. The non-abstract ones are in everyday life, such as provinces in Japan, students at USP, among others.

Representation of a set

The representation of sets are given by capital letters such as:

  • A
  • B
  • $\mathbb{Q}$
  • $\mathbb{Z}$
  • $\mathbb{N}$
  • $\mathbb{R}$
  • among others.

The elements of the sets will be enclosed in braces $\\{\\}$. Below we present some examples of sets.

  • $A = \\{ \text{províncias do Japão}\\}$
Example 1.
  • $B = \\{ \text{inteiros ímpares}\\}$
Example 1.
  • $C = \\{ 1,3,5\\}$
Example 1.
  • $D = \mathbb{N} = \\{\text{conjuntos dos números inteiros}\\}$
Example 1.

In the example 1 the most formal way to represent this set is to place the symbol such that, which can be used with the symbols:

  • $|$
    • $A = \\{ x | x\text{ é província do Japão}\\}$
  • $;$
    • $A = \\{ x ; ~ x\text{ é província do Japão}\\}$
  • or writing such that.
    • $A = \\{ x \text{ tal que } x\text{ é província do Japão}\\}$

The translation of the reading of these sets is done as follows:

  • $A$ is the set of $x$ such that $x$ is a province of Japan.

Relevance

In relevance, the relationship between sets can be established as follows:

  • if $x$ is an element and $A$ is a set.
  • the expression $x \in A$ means:
    • “$x$ belongs to $A$”
    • “$x$ is in $A$”

The $x$ described previously is defined as element.

Example relevance

A book belongs to the library sets.

Euclid's work.

Definition (Element): Let be a set $A$ and $x$ such that $x\in A,$ then $x$ is said to be element. More specifically $x$ is an element of $A$ or $x$ belongs to $A$.

Note: the negation of the definition, that is, $x$ is not an element of $A,$ we write as $x \notin A,$ which means “$x$ does not belong to $A$”.

Empty Set

Definition: the set is said to be empty, denoted by $\emptyset$, the set that does not have any elements.

In fact, given any set $A,$ we have

$$\emptyset = \\{x \in A; x \neq x\\}.$$
Euclid's work.

These are the concepts to start studying sets. Wait for the next posts.