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Discrete Maths. 242-213 , Semester 2, 2013 - 2014. Objective to introduce relations, show their connection to sets, and their use in databases. 5. Relations. Overview. Defining a Relation Relations on a Set Properties of a Relation reflexive, symmetric, transitive

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Discrete Maths

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### Discrete Maths

242-213, Semester 2,2013-2014

• Objective

• to introduce relations, show their connection to sets, and their use in databases

5. Relations

### Overview

• Defining a Relation

• Relations on a Set

• Properties of a Relation

• reflexive, symmetric, transitive

• Combining Relations as Sets

• Composition of Relations

• N-ary Relations

• Databases and Relations

• More Information

### 1. Defining a Relation

• We define a relation as the connection between two (or more) sets.

• We need to first define:

• the ordered pair

• the Cartesian Product for two or more sets

### An Ordered Pair

• In an ordered pair of elements a and b, denoted (a, b), the ordering of the entries matters.

• So and iff a = cand b = d

### The Cartesian Product of 2 Sets

• The Cartesian Product of sets A and B is the set consisting of all the ordered pairs (a, b),where :

• One example Cartesian product is the Euclidean plane, which is the set of all ordered pairs of real numbers

• In general

### A Definition of Relation

• A relationR from a set A to a set B is any subset of the Cartesian product A x B.

• Thus, if P = Ax B, , then we say that x is related to y by R and use the notation: xRy

### Relation Example

• Let A={Smith, Johnson} be a set of students

• B={Calc, Discrete Math, History, Programming} is a set of subjects.

• Smith takes Calc and History, while Johnson takes Calc, Discrete Math and Programming.

• Define relation R as: “student xtakes subject y”.

• R={ (Smith, Calc), (Smith, History), (Johnson, Calc), (Johnson, Discrete Math), (Johnson, Programming) }

• So Smith R Calc is true, whileJohnson RHistory is false.

### 2. Relations on a Set

• A relation on the set A is a relation from A to A.

• In other words, a relation on the set A is a subset of AA

• Example: Let A = {1, 2, 3, 4}. Which ordered pairs are in the relation R = {(a, b) | a < b} ?

R = { (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) }

continued

### Graphically:

R = { (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) }

to

1

1

X

X

X

2

2

from

X

X

3

3

X

4

4

### No. of Relations in a Set

• How many different relations can we define on a set A with n elements?

• A relation on a set A is a subset of AA.

• How many elements are in AA?

• There are n2 elements in AA, so how many subsets (== relations on A) does AA have?

• The number of subsets that we can form out of a set with m elements is 2m. Therefore, 2n2 subsets can be formed out of AA.

• Answer: We can define 2n2 different relations on A.

### 3. Properties of a Relation

A relation R on a set Smay have special properties:

• If , is true, then R is called reflexive.

• If is true whenever is true, then R is called symmetric.

• If is true whenever are both true, then R is called transitive.

### Reflexive Examples

• Are the following relations on {1, 2, 3, 4} reflexive?

R = {(1, 1), (1, 2), (2, 3), (3, 3), (4, 4)}

No.

R = {(1, 1), (2, 2), (2, 3), (3, 3), (4, 4)}

Yes.

R = {(1, 1), (2, 2), (3, 3)}

No.

### Transitivity Examples

• Are the following relations on {1, 2, 3, 4} transitive?

R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)}

Yes.

R = {(1, 3), (3, 2), (2, 1)}

No.

R = {(2, 4), (4, 3), (2, 3), (4, 1)}

No.

### 4. Combining Relations as Sets

• Relations are sets, and therefore, we can apply the usual set operations to them.

• If we have two relations R1 and R2, and both from a set A to set B, then we can combine them to obtain R1 R2, R1  R2, or R1 – R2.

• The result us another relation from A to B.

### 5. Composition of Relations

• Let R be a relation from a set A to a set B and S a relation from B to a set C.

• The compositeof R and S is the relation consisting of ordered pairs (a, c), where aA, cC, when there is an element bB such that (a, b)R and (b, c)S.

• We denote the composite of R and S by SR.

• In other words, if relation R contains a pair (a, b) and relation S contains a pair (b, c), then SR contains a pair (a, c).

### Example 1

• Let D and S be relations on A = {1, 2, 3, 4}

D = {(a, b) | b = 5 - a} “b equals (5 – a)”

S = {(a, b) | a < b} “a is smaller than b”

D = {(1, 4), (2, 3), (3, 2), (4, 1)}

S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}

SD = { (2,4), (3,3), (3,4), (4,2), (4,3), (4,4) }

• D maps an element a to the element (5 – a), and afterwards S maps (5 – a) to all elements larger than (5 – a), resulting in S°D = {(a,b) | b > 5 – a} or S°D = {(a,b) | a + b > 5}.

### Example 2

• Let X and Y be relations on A = {1, 2, 3, …}

X = {(a, b) | b = a + 1} “b equals a plus 1”

Y = {(a, b) | b = 3a} “b equals 3 times a”

X = {(1, 2), (2, 3), (3, 4), (4, 5), …}

Y = {(1, 3), (2, 6), (3, 9), (4, 12), …}

XY = { (1,4), (2,7), (3,10), (4, 13), ... }

• Y maps an element a to 3a, and afterwards X maps 3a to 3a + 1

XY = {(a,b) | b = 3a + 1}

### 6. N-ary Relations

• To apply relations to databases, we generalize binary relations into n-ary relations.

• Let A1, A2, …, An be sets. An n-ary relation on these sets is a subset of A1 x A2 x … x An.

• The sets A1, A2, …, An are called the domains of the relation, and n is called its degree.

### Example

• Let R = {(a, b, c) | (a = 2b) (b = 2c), with a, b, c  Z}

• What is the degree of R?

• The degree of R is 3, so its elements are triples

• What are its domains?

• Its domains are all the set of integers

• Is (2, 4, 8) in R?No

• Is (4, 2, 1) in R?Yes

### 7. Databases and Relations

• The database representation based on relations is called therelational data model

• A database consists of n-tuples called records, which are made up of fields

• These fields are the entriesof the n-tuples

• The relational data model represents a database as an n-ary relation

• in other words, a database is a set of records

### Example

• Consider a database of students, whose records are represented as 4-tuples with the fields Student Name, ID Number, Department, and GPA:

R = { (Ackermann, 00231455, CoE, 3.88),(Adams, 00888323, Physics, 3.45),(Chou, 00102147, CoE, 3.79),(Goodfriend, 00453876, Math, 3.45),(Rao, 00678543, Math, 3.90),(Stevens, 00786576, Psych, 2.99) }

• Relations that represent databases are also called tables, since they are often displayed as tables.

### Primary Key

• A domain of an n-ary relation is called a primary key if the n-tuples are uniquely determined by the values from this domain.

• this means that no two records have the same primary key value

• e.g., which of the fieldsStudent Name, ID Number, Department, and GPAare primary keys?

• Student Name and ID Number are primary keys, because no two students have identical values in those fields

• In a real student database, only ID Number would be a primary key

### 8. More Information

• Discrete Mathematics and its ApplicationsKenneth H. RosenMcGraw Hill, 2007, 7th edition

• chapter 9, sections 9.1 – 9.2