## 3.1Set theory

What is a set? By a set $$S$$ we are to understand, according to Georg Cantor, “a collection into a whole, of definite, well-distinguished objects (called the ‘elements’ of $$S$$) of our perception or of our thought.”14

A set is a collection of elements, without repetition and unordered. $$A$$ is a subset of the set $$B$$ if the elements of $$A$$ are also elements of $$B$$; $$B$$ is therefore a superset of $$A$$. point out that formally there is no definition for set, subset, element and membership, as these are considered primitive notions . Due to the variety of possible notations, the ISO 8000-2:2009 notation was chosen.

Let $$A$$ be a set and $$a$$ be an element of $$A$$. $$a \in A$$ symbolizes that $$a$$ is an element of $$A$$. If an element $$b$$ is not an element of $$A$$, write $$b \notin A$$. A set $$A$$ is a subset of the set $$B$$ if all elements of $$A$$ are also elements of $$B$$, symbolized by $$A \subset B$$ or $$B \supset A$$.

Example 3.4 Consider the sets $$T$$, $$M$$ and $$F$$ defined in Example 3.5.

Set-set Element-set
$$M \subset T$$ $$a \in T$$
$$F \subset T$$ $$a \in M$$
$$T \not\subset M$$ $$a \notin F$$
$$T \not\subset F$$ $$e \in T$$
$$F \not\subset M$$ $$e \notin M$$
$$M \not\subset F$$ $$e \in F$$

Definition 3.1 The cardinal of a set indicates its number of elements, denoted by $$|A|$$.

Example 3.5 (Set, subset, element and cardinal) Suppose the set $$T$$ formed by the students who participate in the university’s truco selection. You can note $T = \lbrace Aaron, Beatriz, Carlos, Denivaldo, Evelise \rbrace \equiv \lbrace a,b,c,d,e \rbrace.$ Each student player in the truco selection is a member of $$T$$. The set $$T$$ can be divided into two subsets, $M = \lbrace a,c,d \rbrace$ and $F= \lbrace b,e \rbrace.$ The boys are elements of $$M$$, the girls are elements of $$F$$, $$|T|=5$$, $$|M|=3$$ and $$|F|=2$$.

### 3.1.1 Empty set

Empty set is a set with no elements, of zero cardinality and denoted by $$\lbrace \rbrace$$ or $$\emptyset$$. It usually indicates the result of an intersection operation (Section 3.1.2.2) where there are no elements in common between the considered sets.

### 3.1.2 Operations

The operations on sets are fundamental in Probability theory. A visual way to represent operations between sets is using the Venn diagram.

#### 3.1.2.1 Union $$\cup$$

The union operation is represented by the symbol $$\cup$$. Indicates that the new generated set must consider all elements of the sets involved in this operation. The union of a collection of sets is annotated by $\begin{equation} \cup_{i=1}^{n} A_i = A_1 \cup A_2 \cup \cdots \cup A_n \tag{3.1} \end{equation}$

Example 3.6 (Union) If $$A=\{1,2,3,4,5\}$$ and $$B=\{2,4,6,8\}$$, $A \cup B = \{1,2,3,4,5,6,8\}$

A <- 1:5
B <- c(2,4,6,8)
base::union(A,B)
##  1 2 3 4 5 6 8
base::union(B,A) # unordered, same as union(A,B)
##  2 4 6 8 1 3 5

#### 3.1.2.2 Intersection $$\cap$$

The intersection operation is represented by the symbol $$\cap$$. Indicates that the new set generated should only consider the elements that are common to the sets involved in this operation. The intersection of a collection of sets is denoted by $\begin{equation} \cap_{i=1}^{n} A_i = A_1 \cap A_2 \cap \cdots \cap A_n = A_1 A_2 \cdots A_n \tag{3.2} \end{equation}$

Example 3.7 (Intersection) Suppose again the sets $$A=\{1,2,3,4,5\}$$ and $$B=\{2,4,6,8\}$$. $A \cap B = \{2,4\}$

A <- 1:5
B <- c(2,4,6,8)
base::intersect(A,B)
##  2 4

#### 3.1.2.3 Complement $$A^C$$

The complement or absolute complement of the set $$A$$ indicates that the new generated set should consider the elements that do not belong to $$A$$. It will be represented by $$A^C=\Omega \backslash A$$, where $$\Omega$$ represents the set of all considered elements. It is associated with the word ‘no’ and can be formally described $\begin{equation} A^C = \{ x \in \Omega : x \notin A \} \tag{3.3} \end{equation}$

#### 3.1.2.4 Difference $$B \backslash A$$

The difference or relative complement between the sets $$B$$ and $$A$$ is denoted by $$B \backslash A$$ or $$B-A$$. It can be read as ‘the elements that are in $$B$$ but not in $$A$$’.

Example 3.8 (Difference) Assume again the sets $$A=\{1,2,3,4,5\}$$ and $$B=\{2,4,6,8\}$$. $A \backslash B = \{1,3,5\}$ $B \backslash A = \{6,8\}$

A <- 1:5
B <- c(2,4,6,8)
base::setdiff(A,B)
##  1 3 5
base::setdiff(B,A)
##  6 8
##### Properties

Let $$A$$ and $$B$$ be two sets in a universe $$\Omega$$.

$\begin{equation} A \cup A^C = \Omega \tag{3.4} \end{equation}$

$\begin{equation} A \cap A^C = \emptyset \tag{3.5} \end{equation}$

$\begin{equation} \emptyset^C = \Omega \tag{3.6} \end{equation}$

$\begin{equation} \Omega^C = \emptyset \tag{3.7} \end{equation}$

$\begin{equation} \text{If} \;\; A \in B, \; \text{then} \;\; B^C \in A^C \tag{3.8} \end{equation}$

$\begin{equation} A \backslash B = A \cap B^C \tag{3.9} \end{equation}$

$\begin{equation} (A \backslash B)^C = A^C \cup B \tag{3.10} \end{equation}$

$\begin{equation} A^C \backslash B^C = B \backslash A \tag{3.11} \end{equation}$

Involution $\begin{equation} (A^C)^C = A \tag{3.12} \end{equation}$

De Morgan’s Laws $\begin{equation} (A \cup B)^C = A^C \cap B^C \tag{3.13} \end{equation}$

$\begin{equation} (A \cap B)^C = A^C \cup B^C \tag{3.14} \end{equation}$

### 3.1.3 Power set

Power set of a set $$A$$ is the set containing all subsets of $$A$$, noted here by $$Po(A)$$ . By definition the empty set $$\emptyset$$ is a subset of $$Po(A)$$. The cardinal of the set of parts is given by $$|Po(A)| = 2^{|A|}$$.

Example 3.9 (Cardinal and power set) Let the set $$A = \left\lbrace -9, 0, 5 \right\rbrace$$. It is known that $|A| = 3,$ $|Po(A)| = 2^{3} = 8$ and $Po(A) = \left\lbrace \emptyset, \left\lbrace -9 \right\rbrace, \left\lbrace 0 \right\rbrace, \left\lbrace 5 \right\rbrace, \left\lbrace -9, 0 \right\rbrace, \left\lbrace -9, 5 \right\rbrace, \left\lbrace 0, 5 \right\rbrace, \left\lbrace -9, 0, 5 \right\rbrace \right\rbrace.$

A <- c(-9,0,5)
length(A)
##  3
(ps <- rje::powerSet(A))
## []
## numeric(0)
##
## []
##  -9
##
## []
##  0
##
## []
##  -9  0
##
## []
##  5
##
## []
##  -9  5
##
## []
##  0 5
##
## []
##  -9  0  5
length(ps)
##  8

### 3.1.4 Disjoint sets and partition

Disjoint sets are those that have no elements in common. Equivalently, their intersection is the empty set. A partition is a grouping of the elements of a set into non-empty subsets such that each element is allocated in only one subset.

Example 3.10 (Disjoint set and partition) If people in a population ($$\Omega$$) are classified into male ($$M$$) and female ($$F$$), notated by $\Omega = \{ M,F \}$ this division forms a possible partition of this population since $M \cup F = \Omega$ $M \cap F = \emptyset.$

### References

Cantor, Georg. 1895. “Beiträge Zur Begründung Der Transfiniten Mengenlehre.” Mathematische Annalen 46 (4): 481–512. https://link.springer.com/content/pdf/10.1007/BF02124929.pdf.
Iezzi, G., and C. Murakami. 1977. Fundamentos de Matemática Elementar 1: Conjuntos, Funções. SP Editora Atual. https://barbosadejesu.files.wordpress.com/2021/09/fundamentos-da-matematica-elementar-1-.pdf.
Kamke, Erich. 1950. Theory of Sets. Courier Corporation.

1. Unter einer ‚Menge‘ verstehen wir jede Zusammenfassung $$M$$ von bestimmten wohlunterschiedenen Objekten $$m$$ unserer Anschauung oder unseres Denkens (welche die ‚Elemente‘ von $$M$$ genannt werden) zu einem Ganzen.↩︎