Lecture 1 - Relations

Relations

Definition

A triple $r=(A,B,R)$, where $A,B$ are sets and $R\subseteq A\times B=\{(a,b)|a\in A,b\in B\}$ is called a binary relation. The set $A$ is called the domain, the set $B$ is called the codomain and the set $R$ is called the graph of the relation $r$. If $A=B$, then the relation $r$ is called homogeneous If $(a,b)\in R$, then we sometimes write $arb$ and we say that a has the relation r to b or a and b are related with respect to the relation r.

Definition

Let $r=(A,B,R)$ be a relation and let $X\subseteq A$. Then the set $r(X)=\{b\in B|\exists x\in X:xrb\}$ is called the relation class of X with respect to r. If $x\in X$, then we denote $r<x>=r(\{x\})=\{b\in B|xrb\}$

Notice that $r(X)={\bigcup }_{x\in X}r<x>$ If A,B are finite sets, then $r=(A,B,R)$ may be represented by a diagram consisting of two sets with elements and connecting arrows. For instance, let $r=(A,B,R)$ where $A=\{1,2,3\}$, $B=\{1,2\}$ and $R=\{(1,1),(1,2),(3,1)\}$ Figure: Diagram of a relation. Also note that $r<1>=\{1,2\}=r(A)$

Examples of relations

a. Let $C$ be the set of all children and let $P$ be the set of all parents. Then we may define the relation $r=(C,P,R)$, where $R=\{(c,p)\in C\times P|c\text{is a child of}p\}$ b. The triple $r=(\mathbb{R},\mathbb{R},R)$, where $R=\{(x,y)\in \mathbb{R}\times \mathbb{R}|x\le y\}$ is a homogeneous relation, called the inequality relation on $\mathbb{R}$. We have $r<1>=[1,\infty )=r([1,2])$ c. There are several examples from Number Theory, such as divisibility on $\mathbb{N}$ or on $\mathbb{Z}$, and Geometry, such as parallelism of lines, perpendicularity of lines, congruence of triangles, similarity of triangles d. Let $A$ and $B$ be two sets. Then the triples $o=(A,B,\varnothing ),u=(A,B,A\times B)$ are relations, called the void relation and the universal relation respectively. e. Let $A$ be a set. Then the triple ${\delta }_A=(A,A,{\Delta }_A)$, where ${\Delta }_A=\{(a,a)|a\in A\}$ is called the equality relation on $A$. f. Every function called is a relation. Indeed, a function $f:A\to B$ is determined by its domain $A$, its codomain $B$ and its graph $G_f=\{(x,y)\in A\times B|y=f(x)\}$ Then the triple $(A,B,G_f)$ is a relation. g. Every directed graph is a relation. For instance the directed graph may be seen as a relation $(A,A,R)$, where $A=\{1,2,3,4\}$ and $R=\{(1,2),(1,3),(2,4),(3,4),(4,1)\}$

Functions

Definition

A relation $r=(A,B,R)$ is called a function if $\forall a\in A,|r<a>|=1$ that is, the relation class with respect to r of every $a\in A$ consist of exactly one element.

In what follows, if $f=(A,B,F)$ is a function, we will mainly use the classical notation for a function, namely $f:A\to B$ or sometimes $A\stackrel{f}{\to }B$. The unique element of the set $f<a>$ will be denoted by $f(a)$. Then we have $(a,b)\in F⟺f(a)=b$

Related notions

From relations we get the following notions.

Definition

Let $f:A\to B$ be a function. Then A is called the domain, $B$ is called the codomain and $F=\{(a,f(a))|a\in A\}$ is called the graph of the function f.

Examples of functions and relations

a. b.

Equivalence relations

Recall that a relation $r=(A,B,R)$ is called homogeneous if $A=B$.

Definition

A homogeneous relation $r=(A,A,R)$ on $A$ is called :

1. reflexive (r) if: $\forall x\in A,xrx$
2. transitive (t) if: $x,y,x\in A,xry\text{and}yrz⟹xrz$
3. symmetric (s) if: $x,y\in A,xry⟹yrx$

A homogeneous relation $r=(A,A,R)$ is called an equivalence relation if $r$ has the properties (r), (t), (s).

Examples of equivalence relations

Partitions

Definition

Let $A$ be a non-empty set. Then a family $(A_i{)}_{i\in I}$ of non-empty subsets of $A$ is called a partition of $A$ if:

1. The family $(A_i{)}_{i\in I}$ covers $A$, that is, ${\bigcup }_{i\in I}{A}_i=A$
2. The $A_i$‘s are pairwise disjoint, that is, $i,j\in I,i\ne j⟹A_i\cap A_j=\varnothing$

Examples of partitions

(a) Let $A=1,2,3,4,5$ and $A_1=\{1,2,3\},A_2=\{4\},A_3=\{5\}$. Then $\{A_1,A_2,A_3\}$ is a partition of $A$. (b) Let $A$ be a set. Then $\{\{a\}|a\in A\}$ and $\{A\}$ are partitions of $A$. (c) Let $A_1$ be the set of even integers and $A_2$ the set of odd integers. Then $\{A_1,A_2\}$ is a partition of $\mathbb{Z}$. (d) Consider the intervals $A_n=[n,n+1)$ for every $n\in \mathbb{Z}$. then the family $(A_n{)}_{n\in \mathbb{Z}}$ is a partition of $\mathbb{R}$.

Quotient set