Relations digraphs

Relations & Digraphs

Product SetsDefinition: An ordered pair 𝑎𝑎, 𝑏𝑏 is a listing of the objects/items 𝑎𝑎 and 𝑏𝑏 in a prescribed order: 𝑎𝑎 is the firstand 𝑏𝑏 is the second. (a sequence of length 2)

Definition: The ordered pairs 𝑎𝑎1, 𝑏𝑏1 and 𝑎𝑎2, 𝑏𝑏2 are equal iff 𝑎𝑎1 = 𝑎𝑎2 and 𝑏𝑏1 = 𝑏𝑏2.

Definition: If 𝐴𝐴 and 𝐵𝐵 are two nonempty sets, we define the product set or Cartesian product 𝐴𝐴 × 𝐵𝐵 as the set of all ordered pairs 𝑎𝑎, 𝑏𝑏 with 𝑎𝑎 ∈ 𝐴𝐴 and 𝑏𝑏 ∈ 𝐵𝐵:

𝐴𝐴 × 𝐵𝐵 = 𝑎𝑎, 𝑏𝑏 𝑎𝑎 ∈ 𝐴𝐴 and 𝑏𝑏 ∈ 𝐵𝐵}

© S. Turaev, CSC 1700 Discrete Mathematics 2

Product SetsExample: Let 𝐴𝐴 = 1,2,3 and 𝐵𝐵 = 𝑟𝑟, 𝑠𝑠 , then

𝐴𝐴 × 𝐵𝐵 =

𝐵𝐵 × 𝐴𝐴 =


Product SetsTheorem: For any two finite sets 𝐴𝐴 and 𝐵𝐵,

𝐴𝐴 × 𝐵𝐵 = 𝐴𝐴 ⋅ 𝐵𝐵 .

Proof: Use multiplication principle!


Definitions:

Let 𝐴𝐴 and 𝐵𝐵 be nonempty sets. A relation 𝑅𝑅 from 𝐴𝐴to 𝐵𝐵 is a subset of 𝐴𝐴 × 𝐵𝐵.

If 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐵𝐵 and 𝑎𝑎, 𝑏𝑏 ∈ 𝑅𝑅, we say that 𝑎𝑎 is related to 𝑏𝑏 by 𝑅𝑅, and we write 𝑎𝑎 𝑅𝑅 𝑏𝑏.

If 𝑎𝑎 is not related to 𝑏𝑏 by 𝑅𝑅, we write 𝑎𝑎 𝑅𝑅 𝑏𝑏.

If 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴, we say 𝑅𝑅 is a relation on 𝐴𝐴.



Example 1: Let 𝐴𝐴 = 1,2,3 and 𝐵𝐵 = 𝑟𝑟, 𝑠𝑠 . Then

𝑅𝑅 = 1, 𝑟𝑟 , 2, 𝑠𝑠 , 3, 𝑟𝑟 ⊆ 𝐴𝐴 × 𝐵𝐵

is a relation from 𝐴𝐴 to 𝐵𝐵.

Example 2: Let 𝐴𝐴 and 𝐵𝐵 are sets of positive integer numbers. We define the relation 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐵𝐵 by

𝑎𝑎 𝑅𝑅 𝑏𝑏 ⇔ 𝑎𝑎 = 𝑏𝑏



Example 3: Let 𝐴𝐴 = 1,2,3,4,5 . The relation 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴 is defined by

𝑎𝑎 𝑅𝑅 𝑏𝑏 ⇔ 𝑎𝑎 < 𝑏𝑏

Then 𝑅𝑅 =

Example 4: Let 𝐴𝐴 = 1,2,3,4,5,6,7,8,9,10 . The relation 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴 is defined by

𝑎𝑎 𝑅𝑅 𝑏𝑏 ⇔ 𝑎𝑎|𝑏𝑏

Then 𝑅𝑅 =



Definition: Let 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐵𝐵 be a relation from 𝐴𝐴 to 𝐵𝐵.

The domain of 𝑅𝑅, denoted by Dom 𝑅𝑅 , is the set of elements in 𝐴𝐴 that are related to some element in 𝐵𝐵.

The range of 𝑅𝑅, denoted by Ran 𝑅𝑅 , is the set of elements in 𝐵𝐵 that are second elements of pairs in 𝑅𝑅.



Relations & DigraphsExample 5: Let 𝐴𝐴 = 1,2,3 and 𝐵𝐵 = 𝑟𝑟, 𝑠𝑠 .

𝑅𝑅 = 1, 𝑟𝑟 , 2, 𝑠𝑠 , 3, 𝑟𝑟

Dom R =

Ran R =

Example 6: Let 𝐴𝐴 = 1,2,3,4,5 . The relation 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴 is defined by 𝑎𝑎 𝑅𝑅 𝑏𝑏 ⇔ 𝑎𝑎 < 𝑏𝑏

Dom R =

Ran R =


The Matrix of a RelationDefinition: Let 𝐴𝐴 = 𝑎𝑎1,𝑎𝑎2, … , 𝑎𝑎𝑚𝑚 , 𝐵𝐵 = 𝑏𝑏1, 𝑏𝑏2, … , 𝑏𝑏𝑛𝑛and 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐵𝐵 be a relation. We represent 𝑅𝑅 by the 𝑚𝑚 ×𝑛𝑛 matrix 𝐌𝐌𝑅𝑅 = [𝑚𝑚𝑖𝑖𝑖𝑖], which is defined by

𝑚𝑚𝑖𝑖𝑖𝑖 = �1, 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑖𝑖 ∈ 𝑅𝑅0, 𝑎𝑎𝑖𝑖 , 𝑏𝑏𝑖𝑖 ∉ 𝑅𝑅

The matrix 𝐌𝐌𝑅𝑅 is called the matrix of 𝑅𝑅.

Example: Let 𝐴𝐴 = 1,2,3 and 𝐵𝐵 = 𝑟𝑟, 𝑠𝑠 .

𝑅𝑅 = 1, 𝑟𝑟 , 2, 𝑠𝑠 , 3, 𝑟𝑟 𝐌𝐌𝑅𝑅 =


The Digraph of a RelationDefinition: If 𝐴𝐴 is finite and 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐴𝐴 is a relation. We represent 𝑅𝑅 pictorially as follows:

Draw a small circle, called a vertex/node, for each element of 𝐴𝐴 and label the circle with the corresponding element of 𝐴𝐴.

Draw an arrow, called an edge, from vertex 𝑎𝑎𝑖𝑖 to vertex 𝑎𝑎𝑖𝑖 iff 𝑎𝑎𝑖𝑖 𝑅𝑅 𝑎𝑎𝑖𝑖.

The resulting pictorial representation of 𝑅𝑅 is called a directed graph or digraph of 𝑅𝑅.


The Digraph of a RelationExample: Let 𝐴𝐴 = 1, 2, 3, 4 and

𝑅𝑅 = 1,1 , 1,2 , 2,1 , 2,2 , 2,3 , 2,4 , 3,4 , 4,1

The digraph of 𝑅𝑅:

Example: Let 𝐴𝐴 = 1, 2, 3, 4 and

Find the relation 𝑅𝑅:© S. Turaev, CSC 1700 Discrete Mathematics

1

2

3

4

12

The Digraph of a RelationDefinition: If 𝑅𝑅 is a relation on a set 𝐴𝐴 and 𝑎𝑎 ∈ 𝐴𝐴, then

the in-degree of 𝑎𝑎 is the number of 𝑏𝑏 ∈ 𝐴𝐴 such that 𝑏𝑏, 𝑎𝑎 ∈ 𝑅𝑅;

the out-degree of 𝑎𝑎 is the number of 𝑏𝑏 ∈ 𝐴𝐴 such that 𝑎𝑎, 𝑏𝑏 ∈ 𝑅𝑅.

Example: Consider the digraph:

List in-degrees and out-degrees of all vertices.

© S. Turaev, CSC 1700 Discrete Mathematics

1

2

3

4

13

The Digraph of a RelationExample: Let 𝐴𝐴 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐,𝑑𝑑 and let 𝑅𝑅 be the relation on 𝐴𝐴 that has the matrix

𝐌𝐌𝑅𝑅 =1 00 1

0 00 0

1 10 1

1 00 1

Construct the digraph of 𝑅𝑅 and list in-degrees and out-degrees of all vertices.


The Digraph of a RelationExample: Let 𝐴𝐴 = 1,4,5 and let 𝑅𝑅 be given the digraph

Find 𝐌𝐌𝑅𝑅 and 𝑅𝑅.


1 4

5

15

Paths in Relations & DigraphsDefinition: Suppose that 𝑅𝑅 is a relation on a set 𝐴𝐴. A path of length 𝑛𝑛 in 𝑅𝑅 from 𝑎𝑎 to 𝑏𝑏 is a finite sequence

𝜋𝜋 ∶ 𝑎𝑎, 𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑛𝑛−1, 𝑏𝑏

beginning with 𝑎𝑎 and ending with 𝑏𝑏, such that

𝑎𝑎 𝑅𝑅 𝑥𝑥1, 𝑥𝑥1 𝑅𝑅 𝑥𝑥2, … , 𝑥𝑥𝑛𝑛−1 𝑅𝑅 𝑏𝑏.

Definition: A path that begins and ends at the same vertex is called a cycle:

𝜋𝜋 ∶ 𝑎𝑎, 𝑥𝑥1, 𝑥𝑥2, … , 𝑥𝑥𝑛𝑛−1, 𝑎𝑎


Paths in Relations & DigraphsExample: Give the examples for paths of length 1,2,3,4and 5.


1 2

43

5

17

Paths in Relations & DigraphsDefinition: If 𝑛𝑛 is a fixed number, we define a relation 𝑅𝑅𝑛𝑛as follows: 𝑥𝑥 𝑅𝑅𝑛𝑛 𝑦𝑦 means that there is a path of length 𝑛𝑛from 𝑥𝑥 to 𝑦𝑦.

Definition: We define a relation 𝑅𝑅∞ (connectivity relation for 𝑅𝑅) on 𝐴𝐴 by letting 𝑥𝑥 𝑅𝑅∞ 𝑦𝑦 mean that there is some path from 𝑥𝑥 to 𝑦𝑦.

Example: Let 𝐴𝐴 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐,𝑑𝑑, 𝑒𝑒 and

𝑅𝑅 = 𝑎𝑎, 𝑎𝑎 , 𝑎𝑎, 𝑏𝑏 , 𝑏𝑏, 𝑐𝑐 , 𝑐𝑐, 𝑒𝑒 , 𝑐𝑐,𝑑𝑑 , 𝑑𝑑, 𝑒𝑒 .

Compute (a) 𝑅𝑅2; (b) 𝑅𝑅3; (c) 𝑅𝑅∞.


Paths in Relations & DigraphsLet 𝑅𝑅 be a relation on a finite set 𝐴𝐴 = 𝑎𝑎1, 𝑎𝑎2, … ,𝑎𝑎𝑛𝑛 , and let 𝐌𝐌𝑅𝑅 be the 𝑛𝑛 × 𝑛𝑛 matrix representing 𝑅𝑅.

Theorem 1: If 𝑅𝑅 is a relation on 𝐴𝐴 = 𝑎𝑎1, 𝑎𝑎2, … ,𝑎𝑎𝑛𝑛 , then

𝐌𝐌𝑅𝑅2 = 𝐌𝐌𝑅𝑅 ⊙𝐌𝐌𝑅𝑅 .

Example: Let 𝐴𝐴 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐,𝑑𝑑, 𝑒𝑒 and



Paths in Relations & DigraphsExample: Let 𝐴𝐴 = 𝑎𝑎, 𝑏𝑏, 𝑐𝑐,𝑑𝑑, 𝑒𝑒 and


𝐌𝐌𝑅𝑅 =

1 10 0

0 01 0

00

000

000

000

100

110

Compute 𝐌𝐌𝑅𝑅2.


Reflexive & Irreflexive RelationsDefinition:

A relation 𝑅𝑅 on a set 𝐴𝐴 is reflexive if 𝑎𝑎,𝑎𝑎 ∈ 𝑅𝑅 for all 𝑎𝑎 ∈ 𝐴𝐴, i.e., if 𝑎𝑎 𝑅𝑅 𝑎𝑎 for all 𝑎𝑎 ∈ 𝐴𝐴.

A relation 𝑅𝑅 on a set 𝐴𝐴 is irreflexive if 𝑎𝑎 𝑅𝑅 𝑎𝑎 for all 𝑎𝑎 ∈ 𝐴𝐴.

Example:

Δ = 𝑎𝑎,𝑎𝑎 | 𝑎𝑎 ∈ 𝐴𝐴 , the relation of equality on the set 𝐴𝐴.

𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴| 𝑎𝑎 ≠ 𝑏𝑏 , the relation of inequality on the set 𝐴𝐴.


Reflexive & Irreflexive RelationsExercise: Let 𝐴𝐴 = 1, 2, 3 , and let 𝑅𝑅 = 1,1 , 1,2 . Is 𝑅𝑅 reflexive or irreflexive?

Exercise: How is a reflexive or irreflexive relation identified by its matrix?

Exercise: How is a reflexive or irreflexive relation characterized by the digraph?


(A-, Anti-) Symmetric RelationsDefinition:

A relation 𝑅𝑅 on a set 𝐴𝐴 is symmetric if whenever 𝑎𝑎 𝑅𝑅 𝑏𝑏, then 𝑏𝑏 𝑅𝑅 𝑎𝑎.

A relation 𝑅𝑅 on a set 𝐴𝐴 is asymmetric if whenever 𝑎𝑎 𝑅𝑅 𝑏𝑏, then 𝑏𝑏 𝑅𝑅 𝑎𝑎.

A relation 𝑅𝑅 on a set 𝐴𝐴 is antisymmetric if whenever 𝑎𝑎 𝑅𝑅 𝑏𝑏 and 𝑏𝑏 𝑅𝑅 𝑎𝑎, then 𝑎𝑎 = 𝑏𝑏.


(A-, Anti-) Symmetric RelationsExample: Let 𝐴𝐴 = 1, 2, 3, 4, 5, 6 and let

𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴 | 𝑎𝑎 < 𝑏𝑏

Is 𝑅𝑅 symmetric, asymmetric or antisymmetric?

Symmetry:

Asymmetry:

Antisymmetry:


(A-, Anti-) Symmetric RelationsExample: Let 𝐴𝐴 = 1, 2, 3, 4 and let

𝑅𝑅 = 1,2 , 2,2 , 3,4 , 4,1


Example: Let 𝐴𝐴 = ℤ+ and let

𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴 | 𝑎𝑎 divides 𝑏𝑏



(A-, Anti-) Symmetric RelationsExercise: How is a symmetric, asymmetric or antisymmetric relation identified by its matrix?

Exercise: How is a symmetric, asymmetric or antisymmetric relation characterized by the digraph?


Transitive RelationsDefinition: A relation 𝑅𝑅 on a set 𝐴𝐴 is transitive if whenever 𝑎𝑎 𝑅𝑅 𝑏𝑏 and 𝑏𝑏 𝑅𝑅 𝑐𝑐 then 𝑎𝑎 𝑅𝑅 𝑐𝑐.

Example: Let 𝐴𝐴 = 1, 2, 3, 4 and let

𝑅𝑅 = 1,2 , 1,3 , 4,2

Is 𝑅𝑅 transitive?

Example: Let 𝐴𝐴 = ℤ+ and let

𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴 | 𝑎𝑎 divides 𝑏𝑏

Is 𝑅𝑅 transitive? © S. Turaev, CSC 1700 Discrete Mathematics 27

Transitive RelationsExercise: Let 𝐴𝐴 = 1,2,3 and 𝑅𝑅 be the relation on 𝐴𝐴whose matrix is

𝐌𝐌𝑅𝑅 =1 1 10 0 10 0 1

Show that 𝑅𝑅 is transitive. (Hint: Check if 𝐌𝐌𝑅𝑅 ⊙2 = 𝐌𝐌𝑅𝑅)

Exercise: How is a transitive relation identified by its matrix?

Exercise: How is a transitive relation characterized by the digraph?© S. Turaev, CSC 1700 Discrete Mathematics 28

Equivalence RelationsDefinition: A relation 𝑅𝑅 on a set 𝐴𝐴 is called an equi-valence relation if it is reflexive, symmetric and transitive.

Example: Let 𝐴𝐴 = 1, 2, 3, 4 and let

𝑅𝑅 = 1,1 , 1,2 , 2,1 , 2,2 , 3,4 , 4,3 , 3,3 , 4,4 .

Then 𝑅𝑅 is an equivalence relation.

Example: Let 𝐴𝐴 = ℤ and let

𝑅𝑅 = 𝑎𝑎, 𝑏𝑏 ∈ 𝐴𝐴 × 𝐴𝐴 ∶ 𝑎𝑎 ≡ 𝑏𝑏 mod 2 .

Show that 𝑅𝑅 is an equivalence relation.


Exercises : Relations






Education

Relations digraphs