discrete maths
Download
Skip this Video
Download Presentation
Discrete Maths

Loading in 2 Seconds...

play fullscreen
1 / 23

Discrete Maths - PowerPoint PPT Presentation


  • 99 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Discrete Maths' - decima


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
discrete maths
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
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
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
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 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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
example1
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
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
8. More Information
  • Discrete Mathematics and its ApplicationsKenneth H. RosenMcGraw Hill, 2007, 7th edition
    • chapter 9, sections 9.1 – 9.2
ad