Relations and their Properties
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 9, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Closures of Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 20, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Relations and their Properties
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 9, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Closures of Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 20, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Representing Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 14, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 1/ Theory of Cartesian product Relation in a Set Types of Relations Equivalence Relation and Equivalence Class explained with examples
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 2/ Types of relations/ Reflexive Symmetric and Transitive relations/ Equivalence relation/ Equivalence class
Representing Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 14, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 1/ Theory of Cartesian product Relation in a Set Types of Relations Equivalence Relation and Equivalence Class explained with examples
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 2/ Types of relations/ Reflexive Symmetric and Transitive relations/ Equivalence relation/ Equivalence class
the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown
Discrete Mathematics - Relations. ... Relations may exist between objects of the same set or between objects of two or more sets. Definition and Properties. A binary relation R from set x to y (written as x R y o r R ( x , y ) ) is a subset of the Cartesian product x × y .
These slides cover the relation between two sets, domain and range of a relation, the inverse of a relation, and composition of relations.
Videos explaining these slides are available
Part1: Definition of relations https://youtu.be/MR0ALeKvaSc
Part2: Domain and range of a relation https://youtu.be/kBlUWTDn-Z4
Part3: The inverse of a relation https://youtu.be/Uv5KFusvvsY
Part4: Composition of relations https://youtu.be/IvVIjJqqPH0
Reference for these slides are
A Transition to Advanced Mathematics 8th Edition,
by Douglas Smith, Maurice Eggen, Richard St. Andre. ISBN-13: 978-1285463261, published by Cengage Learning (August 6, 2014).
https://www.cengagebrain.co.uk/shop/i...
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
2. RelationsRelations
When (a, b) belongs to R, a is said to beWhen (a, b) belongs to R, a is said to be relatedrelated to bto b
by R.by R.
Example:Example: Let P be a set of people, C be a set of cars,Let P be a set of people, C be a set of cars,
and D be the relation describing which person drivesand D be the relation describing which person drives
which car(s).which car(s).
P = {Carl, Suzanne, Peter, Carla},P = {Carl, Suzanne, Peter, Carla},
C = {Mercedes, BMW, tricycle}C = {Mercedes, BMW, tricycle}
D = {(Carl, Mercedes), (Suzanne, Mercedes),D = {(Carl, Mercedes), (Suzanne, Mercedes),
(Suzanne, BMW), (Peter, tricycle)}(Suzanne, BMW), (Peter, tricycle)}
This means that Carl drives a Mercedes, SuzanneThis means that Carl drives a Mercedes, Suzanne
drives a Mercedes and a BMW, Peter drives a tricycle,drives a Mercedes and a BMW, Peter drives a tricycle,
and Carla does not drive any of these vehicles.and Carla does not drive any of these vehicles.
3. Functions as RelationsFunctions as Relations
You might remember that aYou might remember that a functionfunction f from a set A tof from a set A to
a set B assigns a unique element of B to eacha set B assigns a unique element of B to each
element of A.element of A.
TheThe graphgraph of f is the set of ordered pairs (a, b) suchof f is the set of ordered pairs (a, b) such
that b = f(a).that b = f(a).
Since the graph of f is a subset of ASince the graph of f is a subset of A××B, it is aB, it is a relationrelation
from A to B.from A to B.
Moreover, for each elementMoreover, for each element aa of A, there is exactlyof A, there is exactly
one ordered pair in the graph that hasone ordered pair in the graph that has aa as its firstas its first
element.element.
4. Functions as RelationsFunctions as Relations
Conversely, if R is a relation from A to B such thatConversely, if R is a relation from A to B such that
every element in A is the first element of exactly oneevery element in A is the first element of exactly one
ordered pair of R, then a function can be defined withordered pair of R, then a function can be defined with
R as its graph.R as its graph.
This is done by assigning to an element aThis is done by assigning to an element a∈∈A theA the
unique element bunique element b∈∈B such that (a, b)B such that (a, b)∈∈R.R.
5. Relations on a SetRelations on a Set
Definition:Definition: A relation on the set A is a relation from AA relation on the set A is a relation from A
to A.to A.
In other words, a relation on the set A is a subset ofIn other words, a relation on the set A is a subset of
AA××A.A.
Example:Example: Let A = {1, 2, 3, 4}. Which ordered pairs areLet A = {1, 2, 3, 4}. Which ordered pairs are
in the relation R = {(a, b) | a < b} ?in the relation R = {(a, b) | a < b} ?
6. Relations on a SetRelations on a Set
Solution:Solution: R = {R = {(1, 2),(1, 2), (1, 3),(1, 3), (1, 4),(1, 4), (2, 3),(2, 3),(2, 4),(2, 4),(3, 4)}(3, 4)}
RR 11 22 33 44
11
22
33
44
11 11
22
33
44
22
33
44
XX XX XX
XX XX
XX
7. Relations on a SetRelations on a Set
How many different relations can we define on aHow many different relations can we define on a
set A with n elements?set A with n elements?
A relation on a set A is a subset of AA relation on a set A is a subset of A××A.A.
How many elements are in AHow many elements are in A××A ?A ?
There are nThere are n22
elements in Aelements in A××A, so how many subsetsA, so how many subsets
(= relations on A) does A(= relations on A) does A××A have?A have?
The number of subsets that we can form out of a setThe number of subsets that we can form out of a set
with m elements is 2with m elements is 2mm
. Therefore, 2. Therefore, 2nn22
subsets can besubsets can be
formed out of Aformed out of A××A.A.
Answer:Answer: We can define 2We can define 2nn22
different relationsdifferent relations
on A.on A.
8. Properties of RelationsProperties of Relations
We will now look at some useful ways to classifyWe will now look at some useful ways to classify
relations.relations.
Definition:Definition: A relation R on a set A is calledA relation R on a set A is called reflexivereflexive
if (a, a)if (a, a)∈∈R for every element aR for every element a∈∈A.A.
Are the following relations on {1, 2, 3, 4} reflexive?Are the following relations on {1, 2, 3, 4} reflexive?
R = {(1, 1), (1, 2), (2, 3), (3, 3), (4, 4)}R = {(1, 1), (1, 2), (2, 3), (3, 3), (4, 4)} No.No.
R = {(1, 1), (2, 2), (2, 3), (3, 3), (4, 4)}R = {(1, 1), (2, 2), (2, 3), (3, 3), (4, 4)} Yes.Yes.
R = {(1, 1), (2, 2), (3, 3)}R = {(1, 1), (2, 2), (3, 3)} No.No.
Definition:Definition: A relation on a set A is calledA relation on a set A is called irreflexiveirreflexive ifif
(a, a)(a, a)∉∉R for every element aR for every element a∈∈A.A.
9. Properties of RelationsProperties of Relations
Definitions:Definitions:
A relation R on a set A is calledA relation R on a set A is called symmetricsymmetric if (b, a)if (b, a)∈∈RR
whenever (a, b)whenever (a, b)∈∈R for all a, bR for all a, b∈∈A.A.
A relation R on a set A is calledA relation R on a set A is called antisymmetricantisymmetric ifif
a = b whenever (a, b)a = b whenever (a, b)∈∈R and (b, a)R and (b, a)∈∈R.R.
A relation R on a set A is calledA relation R on a set A is called asymmetricasymmetric ifif
(a, b)(a, b)∈∈R implies that (b, a)R implies that (b, a)∉∉R for all a, bR for all a, b∈∈A.A.
10. Properties of RelationsProperties of Relations
Are the following relations on {1, 2, 3, 4}Are the following relations on {1, 2, 3, 4}
symmetric, antisymmetric, or asymmetric?symmetric, antisymmetric, or asymmetric?
R = {(1, 1), (1, 2), (2, 1), (3, 3), (4, 4)}R = {(1, 1), (1, 2), (2, 1), (3, 3), (4, 4)} symmetricsymmetric
R = {(1, 1)}R = {(1, 1)} sym. andsym. and
antisym.antisym.
R = {(1, 3), (3, 2), (2, 1)}R = {(1, 3), (3, 2), (2, 1)} antisym.antisym.
and asym.and asym.
R = {(4, 4), (3, 3), (1, 4)}R = {(4, 4), (3, 3), (1, 4)} antisym.antisym.
11. Properties of RelationsProperties of Relations
Definition:Definition: A relation R on a set A is calledA relation R on a set A is called transitivetransitive
if whenever (a, b)if whenever (a, b)∈∈R and (b, c)R and (b, c)∈∈R, then (a, c)R, then (a, c)∈∈R forR for
a, b, ca, b, c∈∈A.A.
Are the following relations on {1, 2, 3, 4}Are the following relations on {1, 2, 3, 4}
transitive?transitive?
R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)}R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)} Yes.Yes.
R = {(1, 3), (3, 2), (2, 1)}R = {(1, 3), (3, 2), (2, 1)} No.No.
R = {(2, 4), (4, 3), (2, 3), (4, 1)}R = {(2, 4), (4, 3), (2, 3), (4, 1)} No.No.
12. Counting RelationsCounting Relations
Example:Example: How many different reflexive relations canHow many different reflexive relations can
be defined on a set A containing n elements?be defined on a set A containing n elements?
Solution:Solution: Relations on R are subsets of ARelations on R are subsets of A××A, whichA, which
contains ncontains n22
elements.elements.
Therefore, different relations on A can be generatedTherefore, different relations on A can be generated
by choosing different subsets out of these nby choosing different subsets out of these n22
elements, so there are 2elements, so there are 2nn22
relations.relations.
AA reflexivereflexive relation, however,relation, however, mustmust contain the ncontain the n
elements (a, a) for every aelements (a, a) for every a∈∈A.A.
Consequently, we can only choose among nConsequently, we can only choose among n22
– n =– n =
n(n – 1) elements to generate reflexive relations, son(n – 1) elements to generate reflexive relations, so
there are 2there are 2n(n – 1)n(n – 1)
of them.of them.
13. Combining RelationsCombining Relations
Relations are sets, and therefore, we can apply theRelations are sets, and therefore, we can apply the
usualusual set operationsset operations to them.to them.
If we have two relations RIf we have two relations R11 and Rand R22, and both of them, and both of them
are from a set A to a set B, then we can combineare from a set A to a set B, then we can combine
them to Rthem to R11 ∪∪ RR22, R, R11 ∩∩ RR22, or R, or R11 – R– R22..
In each case, the result will beIn each case, the result will be another relation fromanother relation from
A to BA to B..
14. Combining RelationsCombining Relations
…… and there is another important way to combineand there is another important way to combine
relations.relations.
Definition:Definition: Let R be a relation from a set A to a set BLet R be a relation from a set A to a set B
and S a relation from B to a set C. Theand S a relation from B to a set C. The compositecomposite ofof
R and S is the relation consisting of ordered pairs (a,R and S is the relation consisting of ordered pairs (a,
c), where ac), where a∈∈A, cA, c∈∈C, and for which there exists anC, and for which there exists an
element belement b∈∈B such that (a, b)B such that (a, b)∈∈R andR and
(b, c)(b, c)∈∈S. We denote the composite of R and S byS. We denote the composite of R and S by
SS°° RR..
In other words, if relation R contains a pair (a, b) andIn other words, if relation R contains a pair (a, b) and
relation S contains a pair (b, c), then Srelation S contains a pair (b, c), then S°° R contains aR contains a
pair (a, c).pair (a, c).
15. Combining RelationsCombining Relations
Example:Example: Let D and S be relations on A = {1, 2, 3, 4}.Let D and S be relations on A = {1, 2, 3, 4}.
D = {(a, b) | b = 5 - a} “b equals (5 – a)”D = {(a, b) | b = 5 - a} “b equals (5 – a)”
S = {(a, b) | a < b} “a is smaller than b”S = {(a, b) | a < b} “a is smaller than b”
D = {(1, 4), (2, 3), (3, 2), (4, 1)}D = {(1, 4), (2, 3), (3, 2), (4, 1)}
S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}
SS°° D = {D = { (2, 4),(2, 4), (3, 3),(3, 3), (3, 4),(3, 4), (4, 2),(4, 2), (4, 3),(4, 3),
D maps an element a to the element (5 – a), andD maps an element a to the element (5 – a), and
afterwards S maps (5 – a) to all elements larger than (5afterwards S maps (5 – a) to all elements larger than (5
– a), resulting in– a), resulting in SS°°D = {(a,b) | b > 5 – a}D = {(a,b) | b > 5 – a} oror SS°°D = {(a,b)D = {(a,b)
| a + b > 5}.| a + b > 5}.
(4, 4)}(4, 4)}
16. Combining RelationsCombining Relations
We already know thatWe already know that functionsfunctions are justare just
special casesspecial cases ofof relationsrelations (namely those that(namely those that
map each element in the domain onto exactlymap each element in the domain onto exactly
one element in the codomain).one element in the codomain).
If we formally convert two functions intoIf we formally convert two functions into
relations, that is, write them down as sets ofrelations, that is, write them down as sets of
ordered pairs, the composite of these relationsordered pairs, the composite of these relations
will be exactly the same as the composite ofwill be exactly the same as the composite of
the functions (as defined earlier).the functions (as defined earlier).
17. Combining RelationsCombining Relations
Definition:Definition: Let R be a relation on the set A. TheLet R be a relation on the set A. The
powers Rpowers Rnn
, n = 1, 2, 3, …, are defined inductively by, n = 1, 2, 3, …, are defined inductively by
RR11
= R= R
RRn+1n+1
= R= Rnn
°° RR
In other words:In other words:
RRnn
= R= R°° RR°° …… °° R (n times the letter R)R (n times the letter R)
18. Combining RelationsCombining Relations
Theorem:Theorem: The relation R on a set A is transitive if andThe relation R on a set A is transitive if and
only if Ronly if Rnn
⊆⊆ R for all positive integers n.R for all positive integers n.
Remember the definition of transitivity:Remember the definition of transitivity:
Definition:Definition: A relation R on a set A is called transitiveA relation R on a set A is called transitive
if whenever (a, b)if whenever (a, b)∈∈R and (b, c)R and (b, c)∈∈R, then (a, c)R, then (a, c)∈∈R forR for
a, b, ca, b, c∈∈A.A.
The composite of R with itself contains exactly theseThe composite of R with itself contains exactly these
pairs (a, c).pairs (a, c).
Therefore, for a transitive relation R, RTherefore, for a transitive relation R, R°° R does notR does not
contain any pairs that are not in R, so Rcontain any pairs that are not in R, so R°° RR ⊆⊆ R.R.
Since RSince R°° R does not introduce any pairs that are notR does not introduce any pairs that are not
already in R, it must also be true that (Ralready in R, it must also be true that (R°° R)R)°° RR ⊆⊆ R, andR, and
so on, so that Rso on, so that Rnn
⊆⊆ R.R.