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:

  • 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.

Definitions

Definition 1: 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 $A$”.

This pair of sets and membership is the first primitive notion that we must keep in mind when working with sets and elements.

The convention used is the use of uppercase letters for sets and lowercase letters to denote elements.

Next we define the notion of equality of sets.

Definition 2: Let there be two sets $A$ and $B$ such that $A=B$. In this case we say that the set $A$ is equal to $B$ and vice versa. Symbolically we write:

$$A=B \quad \Leftrightarrow \quad (\forall x)~ [(x\in A \rightarrow x\in B) \wedge (x\in B \rightarrow x\in A)]$$

“For all $x$, ($x$ belongs to $A$ implies that $x$ belongs to $B$) and ($x$ belongs to $B$ implies that $x$ belongs to $A$).”

Euclid's work.

In another way, we can say that $A$ is equal to $B$ if and only if $A$ and $B$ have the same elements.

Next we present the first axiom for set theory.

Extension Axiom

Extension Axiom: If $x=y$ and $x\in A,$ then $y\in A$.

Inclusion

In set theory, the notion of inclusion is important, as certain sets can be included in others, that is, the notion of subsets.

Euclid's work.

Definition of Inclusion

Definition 3: Let A and B be sets, we say that:

  • $A$ is contained in $B,$ (or $B$ contains $A$)
  • we denote it by $a \subseteq B$ ( or $B \supseteq A$),
  • if every element of $A$ is an element of $B.$
  • In this case, we also say that $A$ is a subset of $B.$

Symbolically we write:

$$A \subseteq B \quad \Leftrightarrow \quad (\forall x) (x\in A \rightarrow x \in B).$$
Euclid's work.

Definition of proper subset

Now let’s define our own subset.

Definition 3: If $A \subseteq B$ and $A$ is different from B, we say that $A$ is a proper subset of $B$ and we denote it by $A \subsetneq B$ or $A \subset B.$

From this, we obtain the first proposition of the post.

Inclusion Proposition

Proposition: Let $A, B$ and $C$ be sets. So we have:

  1. Reflective: $A = A$.
  2. Symmetric: $A = B \Rightarrow B = A$.
  3. Transitive: $A = B$ and $B = C \Rightarrow A = C$.
  4. Reflective: $A \subseteq A$.
  5. Anti-symmetric: $A \subseteq B$ and $B \subseteq A \Leftrightarrow A = B$
  6. Transitive: $A \subseteq B$ and $B \subseteq C \Rightarrow A \subseteq C$.

Demo:

As an illustration, we will demonstrate some items, the rest will remain as an exercise.

  1. The proposition $x \in A \rightarrow x \in A$ is a tautology,

From definition 1, we have $A = A.$

  1. From the definition 1 we have
$$A = B \Leftrightarrow (\forall x)[(x \in A \rightarrow x ∈ B) \wedge (x \in B \rightarrow x \in A)$$

Now, by the commutativity of the connective $\wedge$ and the definition 1, we obtain that $B = A.$

  1. From the definition 2, we have that $A \subseteq B \wedge B \subseteq A$ is equivalent to the proposition
$$(\forall x) [(x \in A \rightarrow x \in B) \wedge (x \in B \rightarrow x \in A)]$$

which in turn is equivalent to $A = B$ by the definition 1. $\square$

So far we have specified some primitive concepts of set theory. Wait for the next posts.