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Chapter 3: Relations and Posets. Discrete Mathematical Structures: Theory and Applications. Learning Objectives. Learn about relations and their basic properties Explore equivalence relations Become aware of closures Learn about posets

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Chapter 3: Relations and Posets

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## Chapter 3: Relations and Posets

Discrete Mathematical Structures: Theory and Applications

### Learning Objectives

• Learn about relations and their basic properties

• Explore equivalence relations

• Become aware of closures

• Explore how relations are used in the design of relational databases

Discrete Mathematical Structures: Theory and Applications

### Relations

• Relations are a natural way to associate objects of various sets

Discrete Mathematical Structures: Theory and Applications

### Relations

• R can be described in

• Roster form

• Set-builder form

Discrete Mathematical Structures: Theory and Applications

### Relations

• Arrow Diagram

• Write the elements of A in one column

• Write the elements B in another column

• Draw an arrow from an element, a, of A to an element, b, of B, if (a ,b) R

• Here, A = {2,3,5} and B = {7,10,12,30} and R from A into B is definedas follows: For all a  A and b  B, a R b if and only if a divides b

• The symbol → (called an arrow) represents the relationR

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

• Directed Graph

• Let R be a relation on a finite set A

• Describe Rpictorially as follows:

• For each element of A , draw a small or big dot and label the dot by the corresponding element of A

• Draw an arrow from a dot labeleda , to another dot labeled, b , ifa R b .

• Resulting pictorial representation ofR iscalled the directed graph representation of the relationR

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

• Directed graph (Digraph) representation of R

• Each dot is called a vertex

• If a vertex is labeled,a, then it is also called vertexa

• An arc from a vertex labeleda, to another vertex,b is called a directed edge, or directed arc froma tob

• The ordered pair(A , R) a directed graph, or digraph, of the relationR, where each element of Ais a called a vertex of the digraph

Discrete Mathematical Structures: Theory and Applications

### Relations

• Directed graph (Digraph) representation of R (Continued)

• For verticesa and b , ifa R b, a is adjacent tob andb is adjacent froma

• Because (a, a) R, an arc from a to a is drawn; because (a, b) R, an arc is drawn from a to b. Similarly, arcs are drawn from b to b, b to c , b to a, b to d, and c to d

• For an element a A such that (a, a) R, a directed edge is drawn from a to a. Such a directed edge is called a loop at vertex a

Discrete Mathematical Structures: Theory and Applications

### Relations

• Directed graph (Digraph) representation of R (Continued)

• Position of each vertex is not important

• In the digraph of a relationR, there is a directed edge or arc from a vertexa toa vertexb if and only ifa R b

• Let A ={a ,b ,c ,d} and let R be the relation defined by the following set:

R = {(a ,a ), (a ,b ), (b ,b ), (b ,c ), (b ,a ), (b ,d ), (c ,d )}

Discrete Mathematical Structures: Theory and Applications

### Relations

• Domain and Range of the Relation

• Let R be a relation from a set A into a set B. Then R ⊆A x B. The elements of the relation R tell which element of A is R-related to which element of B

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

• Let A = {1, 2, 3, 4} and B = {p, q, r}. Let R = {(1, q), (2, r ), (3, q), (4, p)}. Then R−1= {(q, 1), (r , 2), (q, 3), (p, 4)}

• To find R−1, just reverse the directions of the arrows

• D(R) = {1, 2, 3, 4} = Im(R−1), Im(R) = {p, q, r} = D(R−1)

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

• Constructing New Relations from Existing Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

• Example:

• Consider the relations R and S as given in Figure 3.7.

• The composition S ◦ R is given by Figure 3.8.

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

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Example 3.1.26 continued

Discrete Mathematical Structures: Theory and Applications

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Discrete Mathematical Structures: Theory and Applications

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Example 3.1.27 continued

Discrete Mathematical Structures: Theory and Applications

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Discrete Mathematical Structures: Theory and Applications

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Example 3.1.31 continued

Discrete Mathematical Structures: Theory and Applications

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Discrete Mathematical Structures: Theory and Applications

### Relations

Example 3.1.32 continued

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

• Consider the relation R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 4), (4, 1), (2, 3), (3, 2)} on the set S = {1, 2, 3, 4, 5}.

• The digraph is divided into three distinct blocks. From these blocks the subsets {1, 4}, {2, 3}, and {5} are formed.These subsets are pairwise disjoint, and their union is S.

• A partition of a nonempty set S is a division of S into nonintersecting nonempty subsets.

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

• Example: Let A denote the set of the lowercase English alphabet. Let B be the set of lowercase consonants and C be the set of lowercase vowels. Then B and C are nonempty, B ∩ C = , and A = B ∪ C. Thus, {B, C} is a partition of A.

• Let A be a set and let {A1, A2, A3, A4, A5} be a partition of A. Corresponding to this partition, a Venn diagram, can be drawn, Figure 3.13

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Relations

• Let A = {a, b, c , d, e , f , g , h, i, j }. Let R be a relation on A such that the digraph of R is as shown in Figure 3.14.

• Then a, b, c , d, e , f , c , g is a directed walk in R as a R b,b R c,c R d,d R e, e R f , f R c, c R g. Similarly, a, b, c , g is also a directed walk in R. In the walk a, b, c , d, e , f , c , g , the internal vertices are b, c , d, e , f , and c , which are not distinct as c repeats.

• this walk is not a path. In the walk a, b, c , g , the internal vertices are b and c , which are distinct. Therefore, the walk a, b, c, g is a path.

Discrete Mathematical Structures: Theory and Applications

### Relations

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Digraphs of Posets

• Because any partial order is also a relation, a digraph representation of partial order may be given.

• Example: On the set S = {a, b, c}, consider the relation R = {(a, a), (b, b), (c , c ), (a, b)}.

• From the directed graph it follows that the given relation is reflexive and transitive.

• This relation is also antisymmetric because there is a directed edge from a to b, but there is no directed edge from b to a. Again, in the graph there are two distinct vertices a and c such that there are no directed edges from a to c and from c to a.

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Digraphs of Posets

• Let S = {1, 2, 3, 4, 6, 12}. Consider the divisibility relation on S, which is a partial order

• A digraph of this poset is as shown in Figure 3.20

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Closed Path

• On the set S = {a, b, c } consider the relation R = {(a, a), (b, b), (c , c ), (a, b), (b, c ), (c , a)}

• The digraph of this relation is given in Figure 3.21

• In this digraph, a, b, c , a form a closed path. Hence, the given relation is not a partial order relation

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Hasse Diagram

• Let S = {1, 2, 3}. Then P(S) = {, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, S}

• Now (P(S),≤) is a poset, where ≤ denotes the set inclusion relation. The poset diagram of (P(S),≤) is shown in Figure 3.22

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Hasse Diagram

• Let S = {1, 2, 3}. Then P(S) = {, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, S}

• (P(S),≤) is a poset, where ≤ denotes the set inclusion relation

• Draw the digraph of this inclusion relation (see Figure 3.23). Place the vertex A above vertex B if B ⊂ A. Now follow steps (2), (3), and (4)

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Hasse Diagram

• Consider the poset (S,≤), where S = {2, 4, 5, 10, 15, 20} and the partial order ≤ is the divisibility relation

• In this poset, there is no element b ∈ S such that b  5 and b divides 5. (That is, 5 is not divisible by any other element of S except 5). Hence, 5 is a minimal element. Similarly, 2 is a minimal element

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Hasse Diagram

• 10 is not a minimal element because 2 ∈ S and 2 divides 10. That is, there exists an element b ∈ S such that b < 10. Similarly, 4, 15, and 20 are not minimal elements

• 2 and 5 are the only minimal elements of this poset. Notice that 2 does not divide 5. Therefore, it is not true that 2 ≤ b, for all b ∈ S, and so 2 is not a least element in (S,≤). Similarly, 5 is not a least element. This poset has no least element

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Figure 3.24

• Hasse Diagram

• There is no element b ∈ S such that b 15, b > 15, and 15 divides b. That is, there is no element b ∈ S such that 15 < b. Thus, 15 is a maximal element. Similarly, 20 is a maximal element.

• 10 is not a maximal element because 20 ∈ S and 10 divides 20. That is, there exists an element b ∈ S such that 10 < b. Similarly, 4 is not a maximal element.

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Figure 3.24

• Hasse Diagram

• 20 and 15 are the only maximal elements of this poset

• 10 does not divide 15, hence it is not true that b ≤ 15, for all b ∈ S, and so 15 is not a greatest element in (S,≤)

• This poset has no greatest element

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Non-distributive Lattice

• Because a ∧ (b ∨ c ) = a ∧ 1 = a = 0 = 0 ∨ 0 = (a ∧ b) ∨ (a ∧ c ), this is not a distributive lattice

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Complement

• Let D30 denote the set of all positive divisors of 30. Then D30= {1, 2, 3, 5, 6, 10, 15, 30}.

• (D30,≤) is a poset where a ≤ b if and only if a divides b (≤ is the divisibility relation)

• D30,≤) is a poset where a ≤ b if and only if a divides b (≤ is the divisibility relation). Because 1 divides all elements of D30, it follows that 1 ≤ m, for all m ∈ D30. Therefore, 1 is the least element of this poset.

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Complement

• Let a, b ∈ D30. Let d = gcd{a, b} and m = lcm{a, b}. Now d | a and d | b. Hence, d ≤ a and d ≤ b.

• This shows that d is a lower bound of {a, b}. Let c ∈ D30 and c ≤ a, c ≤ b.

• This shows that d is a lower bound of {a, b}. Let c ∈ D30 and c ≤ a, c ≤ b. Then, c | a and c | b and because d = gcd{a, b}, it follows that c | d, so c ≤ d. Thus, d = glb{a, b}.

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Complement

• Because all the positive divisors of a, b are also divisors of 30, d ∈ D30, so d = a ∧ b

• It can be shown that m ∈ D30 and m = a ∨ b

• Hence, (D30,≤) is a lattice with the least element integer 1 and the greatest element 30

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

• Complement

• For any element a ∈ D30, (30/a) ∈ D30

• Using the properties of gcd and lcm, it can be shown that a ∧ (30/a)= 1 and a ∨ (30/a)= 30

• 10 ∧ (30/10) = gcd{10, 3} = 1 and 10 ∨ (30/10) = lcm{10, 3} = 30

• Hence, 3 is a complement of 10 in this lattice

Discrete Mathematical Structures: Theory and Applications

### Partially Ordered Sets

Discrete Mathematical Structures: Theory and Applications

### Application: Relational Database

• A database is a shared and integrated computer structure that stores

• End-user data; i.e., raw facts that are of interest to the end user;

• A database can be thought of as a well-organized electronic file cabinet whose contents are managed by software known as a database management system; that is, a collection of programs to manage the data and control the accessibility of the data

Discrete Mathematical Structures: Theory and Applications

### Application: Relational Database

• In a relational database system, tables are considered as relations

• A table is an n-ary relation, where n is the number of columns in the tables

• The headings of the columns of a table are called attributes, or fields, and each row is called a record

• The domain of a field is the set of all (possible) elements in that column

Discrete Mathematical Structures: Theory and Applications

### Application: Relational Database

• Each entry in the ID column uniquely identifies the row containing that ID

• Such a field is called a primary key

• Sometimes, a primary key may consist of more than one field

Discrete Mathematical Structures: Theory and Applications

### Application: Relational Database

• Structured Query Language (SQL)

• Information from a database is retrieved via a query, which is a request to the database for some information

• A relational database management system provides a standard language, called structured query language (SQL)

• A relational database management system provides a standard language, called structured query language (SQL)

Discrete Mathematical Structures: Theory and Applications

### Application: Relational Database

• Structured Query Language (SQL)

• An SQL contains commands to create tables, insert data into tables, update tables, delete tables, etc.

• Once the tables are created, commands can be used to manipulate data into those tables.

• The most commonly used command for this purpose is the select command. The select command allows the user to do the following:

• Specify what information is to be retrieved and from which tables.

• Specify conditions to retrieve the data in a specific form.

• Specify how the retrieved data are to be displayed.

Discrete Mathematical Structures: Theory and Applications