CSE 311 Lecture 22: Relations and Directed Graphs

Topics

Relations

Relations, properties, operations, and applications.

Directed graphs

Directed graphs and representing relations as directed graphs.

What are relations?

Binary relation

Let $A$ and $B$ be sets.

A binary relation from $A$ to $B$ is a subset of $A\times B$.

A binary relation $R$ is a set of tuples. If $R$ is a relation from a set $A$ to itself, we say that $R$ is a relation on $A$.

Examples of relations we’ve seen before

$\geq$ on $\N$

$\{(x, y) : x\geq y \text{ and } x,y\in\N\}$

$<$ on $\R$

$\{(x, y) : x< y \text{ and } x,y\in\R\}$

$=$ on $\Sigma^* $

$\{(x, y) : x = y \text{ and } x,y\in\Sigma^* \}$

$\subseteq$ on $\mathcal{P}(U)$ for a universe $U$

$\{(A, B) : A \subseteq B \text{ and } A,B\in\mathcal{P}(U) \}$

More examples of relations

$R_1 = \{(a, 1), (a, 2), (b, 1), (b, 3), (c, 3)\}$

$R_2 = \{(x, y) : x \equiv y \, (\text{mod } 5)\}$

$R_3 = \{(c_1, c_2) : c_1 \text{ is a prerequisite for } c_2\}$

$R_4 = \{(s, c) : \text{student } s \text{ has taken course } c\}$

Important properties of relations

Relation properties

Let $R$ be a relation on $A$.

$R$ is reflexive iff $(a,a)\in R$ for every $a\in A$.

$R$ is symmetric iff $(a,b)\in R$ implies $(b,a)\in R$.

$R$ is antisymmetric iff $(a,b)\in R$ and $a\neq b$ implies $(b,a)\not\in R$.

$R$ is transitive iff $(a,b)\in R$ and $(b,c)\in R$ implies $(a,c)\in R$.

Operations on relations

Relation composition

Let $R$ be a relation on $A$ to $B$.

Let $S$ be a relation on $B$ to $C$.

The composition of $R$ and $S$, $R\circ S$, is the relation from $A$ to $C$ defined by:

$R\circ S = \{(a,c) : \exists b. (a,b)\in R \text{ and } (b,c)\in S \}$

Since relations are just sets, we can also combine them with the standard set theoretic operators ($\cup,\cap,\setminus$).

Examples of relational composition

$R\circ S = \{(a,c) : \exists b . (a,b)\in R \text{ and } (b,c)\in S \}$

Consider the relations $\text{Parent}$ and $\text{Sister}$.

$(a, b)\in\text{Parent}$ iff $b$ is a parent of $a$

$(a, b)\in\text{Sister}$ iff $b$ is a sister of $a$

When is $(x, y)\in \text{Parent}\circ\text{Sister}$?

$y$ is an aunt of $x$.

When is $(x, y)\in \text{Sister}\circ\text{Parent}$?

$y$ is a parent of $x$ and $x$ has a sister.

More examples of relational composition

$R\circ S = \{(a,c) : \exists b . (a,b)\in R \text{ and } (b,c)\in S \}$

Use the relations $\text{Parent}, \text{Child}, \text{Sister}, \text{Brother}, \text{Sibling}$ to express

$(a,b)\in\text{Uncle}$ iff $b$ is an uncle of $a$ $\text{Parent}\circ\text{Brother}$

$(a,b)\in\text{Cousin}$ iff $b$ is a cousin of $a$ $\text{Parent}\circ\text{Sibling}\circ\text{Child}$

Powers of a relation

$R^0 = \{(a,a) : a\in A\}$ The identity relation on $A$.

$R^1 = R^0\circ R = R$

$R^2 = R^1\circ R = R \circ R = \{(a,c) : \exists b. (a,b)\in R \text{ and } (b,c)\in R\}$

$\vdots$

$R^{n+1} = R^n \circ R$ for $n \geq 0$

Reflexive transitive closure

Let $R$ be a relation on $A$.

The reflexive transitive closure, $R^* $, of $R$ is defined by

$R^* = \bigcup_{n=0}^\infty R^n = R^0 \cup R^1 \cup R^2 \cup \ldots$

Applications of relations

What is a directed graph?

Directed graphs

A directed graph $G = (V, E)$ consists of a set of vertices $V$ and a set of edges $E\subseteq V\times V$, which are ordered pairs of vertices.

Representing relations as directed graphs

Paths in graphs and relations

Path in a directed graph

Let $G=(V,E)$ be a directed graph.

A path in $G$ is a sequence of vertices $v_0, v_1, \ldots, v_k$ where $(v_i, v_{i+1})\in E$ for each $0\leq i< k$.

We’ve defined paths on graphs but the same definition applies to relations, since a relation and its graph representation are interchangeable.

Relational composition using directed graphs

If $R$ and $S$ are relations on $A$, compute $R\circ S$ as follows:

Create the digraph $G = (A, R\cup S)$.

Add an edge $(a,b)$ to $R\circ S$ iff there is a path $a,v,b$ in $G$ such that $(a,v)\in R$ and $(v, b)\in S$.

Powers of a relation using directed graphs

Reflexive transitive closure using directed graphs