diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index 6ef77f0..96ae5cb 100755 --- a/config/en/mkdocs.yaml +++ b/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 diff --git a/docs/en/mathematics/set-theory/relations.md b/docs/en/mathematics/set-theory/relations.md new file mode 100644 index 0000000..a172a9a --- /dev/null +++ b/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. + +
+ +> *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 $b$. + +
+ +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 + +$$ +\begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix} +$$ + +
+ +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. + +
+ +> *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. + +
+ +> *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 $S/R$. + +
+ +> *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 $S$. + +
\ No newline at end of file diff --git a/docs/en/mathematics/set-theory/sets-and-numbers.md b/docs/en/mathematics/set-theory/sets-and-numbers.md deleted file mode 100755 index cd698ea..0000000 --- a/docs/en/mathematics/set-theory/sets-and-numbers.md +++ /dev/null @@ -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 - -$$ -\mathrm{Let} \ -V = \{x \in \mathbb{R} | 2 \leq x < 4\} = [2 ; 4), \ -W = \{x \in \mathbb{R} | \pi < x < 2\pi\} = (\pi ; 2\pi). -$$ - -$$\mathrm{Interception: } V \cap W = (\pi ; 4).$$ - -$$\mathrm{Union: } V \cup W = [2 ; 2\pi).$$ \ No newline at end of file diff --git a/docs/en/mathematics/set-theory/sets.md b/docs/en/mathematics/set-theory/sets.md index 7ad4554..711dcbf 100644 --- a/docs/en/mathematics/set-theory/sets.md +++ b/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\}
> *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 $\exists$. +
+ > *Proposition* **- DeMorgan's rule**: the statement > > $$