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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 $S$.