Updated sets.md and added part of relations.md

config/en/mkdocs.yaml

@@ -54,6 +54,7 @@ nav:
     - 'Logic': mathematics/logic.md
     - 'Set theory':
       - 'Sets': mathematics/set-theory/sets.md
+      - 'Relations': mathematics/set-theory/relations.md
     - 'Calculus': 
       - 'Limits': mathematics/calculus/limits.md
       - 'Continuity': mathematics/calculus/continuity.md

docs/en/mathematics/set-theory/relations.md

@@ -0,0 +1,80 @@
+# Relations
+
+> *Definition*: a binary relation $R$ between the sets $S$ and $T$ is a subset of the Cartesian product $S \times T$. 
+> 
+> * If $(a,b) \in R$ then $a$ is in relation $R$ to $b$, denoted by $aRb$.
+> * The set $S$ is called the domain of the relation $R$ and the set $T$ the codomain.
+> * If $S=T$ then $R$ is a relation on $S$.
+> * This definition can be expanded to n-ary relations.
+
+<br>
+
+> *Definition*: let $R$ be a relation from a set $S$ to a set $T$. Then for each element $a \in S$ we define $[a]_R$ to be the set
+>
+> $$
+>   [a]_R := \{b \in T \;|\; aRb\}.
+> $$
+>
+> This set is called the ($R$-) image of $a$. 
+>
+> For $b \in T$ the set
+>
+> $$
+>   _R[b] := \{a \in S \;|\; aRb\}
+> $$
+>
+> is called the ($R$-) pre-image of $B$ or $R$-fiber of $b$.
+
+<br>
+
+Relations between finite sets can be described using matrices. 
+
+> *Definition*: if $S = \{s_1, \dots, s_n\}$ and $T = \{t_1, \dots, t_m\}$ are finite sets and $R \subseteq S \times T$ is a binary relation, then the adjacency matrix $A_R$ of the relation $R$ is the $n \times n$ matrix whose rows are indexed by $S$ and columns by $T$ defined by
+>
+> $$
+>   A_{s,t} = \begin{cases} 1 &\text{ if } (s,t) \in R, \\ 0 &\text{ otherwise}. \end{cases}
+> $$
+
+For example, the adjacency matrix of relation $\leq$ on the set $\{1,2,3,4,5\}$ is the upper triangular matrix
+
+$$
+\begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}
+$$
+
+<br>
+
+Some relations have special properties
+
+> *Definitions*: let $R$ be a relation on a set $S$. Then $R$ is called
+>
+> * *Reflexive* if $\forall x \in S$ there is $(x,x) \in R$.
+> * *Irreflexive* if $\forall x \in S$ there is $(x,x) \notin R$.
+> * *Symmetric* if $\forall x,y \in S$ there is that $xRy \implies yRx$.
+> * *Antisymmetric* if $\forall x,y \in S$ there is that $xRy \land yRx \implies x = y$.
+> * *Transitive* if $\forall x,y,z \in S$ there is that $xRy \land yRz \implies xRz$.
+
+## Equivalence relations
+
+> *Definition*: a relation $R$ on a set $S$ is called an equivalence relation on $S$ if and only if it is reflexive, symmetric and transitive.
+
+<br>
+
+> *Lemma*: let $R$ be an equivalence relation on a set $S$. If $b \in [a]_R$, then $[b]_R = [a]_R$.
+
+??? note "*Proof*:"
+
+    Suppose $b \in [a]_R$, therefore $aRb$. If $c \in [b]_R$, then $bRc$ and as $aRb$ there is transitivity $aRc$. In particular $[b]_R \subseteq [a]_R$. By symmetry of $R$, $aRb \implies bRa$ and hence $a \in [b]_R$, obtaining $[a]_R \subseteq [b]_R. 
+
+<br>
+
+> *Definition*: let $R$ be an equivalence relation on a set $S$. Then the sets $[s]_R$ where $s \in S$ are called the $R$-equivalence classes on $S$. The set of $R$-equivalence classes is denoted by $S/R$.
+
+<br>
+
+> *Theorem*: let $R$ be an equivalence relation on a set $S$. Then the set $S/R$ of $R$-equivalence classes partitions the set $S$. 
+
+??? note "*Proof*:"
+
+    Let $\Pi_R$ be the set of $R$-equivalence classes. Then by reflexivity of $R$ we find that each element $a \in S$ is inside the class $[a]_R$ of $\Pi_R$. If an element $a \in S$ is in the classes $[b]_R$ and $[c]_R$ of $\Pi_R$, then by the previous lemma we find $[b]_R = [a]_R$ and $[c]_R = [a]_R$. Then $[b]_R = [c]_R$, therefore each element $a \in S$ is inside a unique member of $\Pi_R$, which therefore is a partition of $S$.
+
+<br>

docs/en/mathematics/set-theory/sets-and-numbers.md

@@ -1,19 +0,0 @@
-# Sets and numbers
-
-A collection of well defined objects. Such as the set of all even numbers
-
-$$
-V = \{x \in \mathbb{N} | x = 2n \space \mathrm{for} \space n \in \mathbb{N}\}.
-$$
-
-## Interception and union
-
-$$
-\mathrm{Let} \
-V = \{x \in \mathbb{R} | 2 \leq x < 4\} = [2 ; 4), \
-W = \{x \in \mathbb{R} | \pi < x < 2\pi\} = (\pi ; 2\pi).
-$$
-
-$$\mathrm{Interception: } V \cap W = (\pi ; 4).$$
-
-$$\mathrm{Union: } V \cup W = [2 ; 2\pi).$$

docs/en/mathematics/set-theory/sets.md

@@ -42,9 +42,11 @@ Suppose for example that $B = {x,y,z}$, then $\wp(B) = \{\varnothing,\{x\},\{y\}
 <br>
 
 > *Definition*: let $P$ be a predicate with reference set $X$, then
+> 
 >$$
 >   \big\{x \in X \;\big|\; P(x) \big\}
 >$$
+>
 > denotes the subset of $X$ consisting of all elements $x \in X$ for which statement $P(x)$ is true.
 
 ## Operations on sets
@@ -132,6 +134,8 @@ For example the set $\{1,2, \dots , 10\}$ can be partitioned into the sets $\{1,
 
 > *Definitions*: the universal quantifier "for all" is denoted by $\forall$ and the existential quantifier "there exists" is denoted by $\exists$.
 
+<br>
+
 > *Proposition* **- DeMorgan's rule**: the statement
 >
 > $$