1 von 31

Ähnlich wie AM103(Unit3).pptx

Sets functions-sequences-exercises

TOUSIF AZIM_BDS 101[1].pptx
TOUSIF AZIM_BDS 101[1].pptxTousifAzim

ARNAV DHAMA.pptx
ARNAV DHAMA.pptxKunal219998

Set Theory, Relations,Functions (Discrete Structures)
Set Theory, Relations,Functions (Discrete Structures)kimdokja738

Discrete mathematics presentation

Discrete mathematics for diploma students
Discrete mathematics for diploma studentsZubair Khan

SetTheory(Dr. Praveen Mittal).pdf
SetTheory(Dr. Praveen Mittal).pdfpraveen mittal

Sets (Mathematics class XI)
Sets (Mathematics class XI)VihaanBhambhani

Maths Project on sets
Maths Project on setsatifansari17

Toc chapter 1 srg
Toc chapter 1 srgShayak Giri

Discrete mathematics OR Structure
Discrete mathematics OR Structure Abdullah Jan

Set Theory - Unit -II (Mathematical Foundation Of Computer Science).pptx
Set Theory - Unit -II (Mathematical Foundation Of Computer Science).pptxKalirajMariappan

Set relationship, set operation and sigmoid
Set relationship, set operation and sigmoidKanza batool

Set-Discrete Mathematics by Arafath.pptx
Set-Discrete Mathematics by Arafath.pptxArafath Islam Sezan

Ähnlich wie AM103(Unit3).pptx(20)

SETS PPT-XI.pptx
SETS PPT-XI.pptx

Sets
Sets

Sets functions-sequences-exercises
Sets functions-sequences-exercises

TOUSIF AZIM_BDS 101[1].pptx
TOUSIF AZIM_BDS 101[1].pptx

ARNAV DHAMA.pptx
ARNAV DHAMA.pptx

Chapter 1, Sets
Chapter 1, Sets

Set Theory, Relations,Functions (Discrete Structures)
Set Theory, Relations,Functions (Discrete Structures)

Discrete mathematics presentation
Discrete mathematics presentation

Discrete mathematics for diploma students
Discrete mathematics for diploma students

Crisp set
Crisp set

SetTheory(Dr. Praveen Mittal).pdf
SetTheory(Dr. Praveen Mittal).pdf

Sets (Mathematics class XI)
Sets (Mathematics class XI)

Maths Project on sets
Maths Project on sets

3a-Set theory.ppt
3a-Set theory.ppt

Toc chapter 1 srg
Toc chapter 1 srg

Discrete mathematics OR Structure
Discrete mathematics OR Structure

Set Theory - Unit -II (Mathematical Foundation Of Computer Science).pptx
Set Theory - Unit -II (Mathematical Foundation Of Computer Science).pptx

Set relationship, set operation and sigmoid
Set relationship, set operation and sigmoid

Fuzzy Logic
Fuzzy Logic

Set-Discrete Mathematics by Arafath.pptx
Set-Discrete Mathematics by Arafath.pptx

Flow Your Strategy at Flight Levels Day 2024
Flow Your Strategy at Flight Levels Day 2024Kirill Klimov

Darshan Hiranandani [News About Next CEO].pdf
Darshan Hiranandani [News About Next CEO].pdfShashank Mehta

Pitch Deck Teardown: Geodesic.Life's \$500k Pre-seed deck
Pitch Deck Teardown: Geodesic.Life's \$500k Pre-seed deckHajeJanKamps

Traction part 2 - EOS Model JAX Bridges.
Traction part 2 - EOS Model JAX Bridges.Anamaria Contreras

Call US-88OO1O2216 Call Girls In Mahipalpur Female Escort Service
Call US-88OO1O2216 Call Girls In Mahipalpur Female Escort Servicecallgirls2057

business environment micro environment macro environment.pptx
business environment micro environment macro environment.pptxShruti Mittal

trending-flavors-and-ingredients-in-salty-snacks-us-2024_Redacted-V2.pdf
trending-flavors-and-ingredients-in-salty-snacks-us-2024_Redacted-V2.pdfMintel Group

8447779800, Low rate Call girls in Saket Delhi NCR
8447779800, Low rate Call girls in Saket Delhi NCRashishs7044

Annual General Meeting Presentation Slides
Annual General Meeting Presentation SlidesKeppelCorporation

Cyber Security Training in Office Environment
Cyber Security Training in Office Environmentelijahj01012

Memorándum de Entendimiento (MoU) entre Codelco y SQM
Memorándum de Entendimiento (MoU) entre Codelco y SQMVoces Mineras

Intermediate Accounting, Volume 2, 13th Canadian Edition by Donald E. Kieso t...
Intermediate Accounting, Volume 2, 13th Canadian Edition by Donald E. Kieso t...ssuserf63bd7

Entrepreneurship lessons in Philippines
Entrepreneurship lessons in PhilippinesDavidSamuel525586

1911 Gold Corporate Presentation Apr 2024.pdf
1911 Gold Corporate Presentation Apr 2024.pdfShaun Heinrichs

Fordham -How effective decision-making is within the IT department - Analysis...
Fordham -How effective decision-making is within the IT department - Analysis...Peter Ward

MAHA Global and IPR: Do Actions Speak Louder Than Words?
MAHA Global and IPR: Do Actions Speak Louder Than Words?Olivia Kresic

Flow Your Strategy at Flight Levels Day 2024
Flow Your Strategy at Flight Levels Day 2024

Darshan Hiranandani [News About Next CEO].pdf
Darshan Hiranandani [News About Next CEO].pdf

Japan IT Week 2024 Brochure by 47Billion (English)
Japan IT Week 2024 Brochure by 47Billion (English)

Pitch Deck Teardown: Geodesic.Life's \$500k Pre-seed deck
Pitch Deck Teardown: Geodesic.Life's \$500k Pre-seed deck

Traction part 2 - EOS Model JAX Bridges.
Traction part 2 - EOS Model JAX Bridges.

Call US-88OO1O2216 Call Girls In Mahipalpur Female Escort Service
Call US-88OO1O2216 Call Girls In Mahipalpur Female Escort Service

business environment micro environment macro environment.pptx
business environment micro environment macro environment.pptx

trending-flavors-and-ingredients-in-salty-snacks-us-2024_Redacted-V2.pdf
trending-flavors-and-ingredients-in-salty-snacks-us-2024_Redacted-V2.pdf

The Bizz Quiz-E-Summit-E-Cell-IITPatna.pptx
The Bizz Quiz-E-Summit-E-Cell-IITPatna.pptx

8447779800, Low rate Call girls in Saket Delhi NCR
8447779800, Low rate Call girls in Saket Delhi NCR

Annual General Meeting Presentation Slides
Annual General Meeting Presentation Slides

Cyber Security Training in Office Environment
Cyber Security Training in Office Environment

Memorándum de Entendimiento (MoU) entre Codelco y SQM
Memorándum de Entendimiento (MoU) entre Codelco y SQM

Intermediate Accounting, Volume 2, 13th Canadian Edition by Donald E. Kieso t...
Intermediate Accounting, Volume 2, 13th Canadian Edition by Donald E. Kieso t...

Entrepreneurship lessons in Philippines
Entrepreneurship lessons in Philippines

1911 Gold Corporate Presentation Apr 2024.pdf
1911 Gold Corporate Presentation Apr 2024.pdf

Fordham -How effective decision-making is within the IT department - Analysis...
Fordham -How effective decision-making is within the IT department - Analysis...

MAHA Global and IPR: Do Actions Speak Louder Than Words?
MAHA Global and IPR: Do Actions Speak Louder Than Words?

Corporate Profile 47Billion Information Technology
Corporate Profile 47Billion Information Technology

AM103(Unit3).pptx

• 1. DISCRETE STRUCTURES AM103 Department of Applied Sciences Chitkara University, Punjab
• 3. Partial Orderings • A partial ordering (or partial order) is a relation that is reflexive, antisymmetric, and transitive – Recall that antisymmetric means that if (a,b)  R, then (b,a) R unless b = a – Thus, (a,a) is allowed to be in R – But since it’s reflexive, all possible (a,a) must be in R 3 Discrete Structure/AM103 Dr. Manoj Gaur
• 4. Partially Ordered Set (POSET) • A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S, R) 4 Discrete Structure/AM103 Dr. Manoj Gaur
• 5. Example (1) • Let S = {1, 2, 3} and • let R = {(1,1), (2,2), (3,3), (1, 2), (3,1), (3,2)} 5 1 2 3 Discrete Structure/AM103 Dr. Manoj Gaur
• 6. Comparable / Incomparable 6 Discrete Structure/AM103 Dr. Manoj Gaur
• 7. Example • Consider the power set of {a, b, c} and the subset relation. (P({a,b,c}), ) 7 { , } { , } and { , } { , } a c a b a b a c   So, {a,c} and {a,b} are incomparable  Discrete Structure/AM103 Dr. Manoj Gaur
• 8. 8 • In the poset (Z+,≤), are the integers 3 and 9 comparable? – Yes, as 3 ≤ 9 • Are 7 and 5 comparable? – Yes, as 5 ≤ 7 • As all pairs of elements in Z+ are comparable, the poset (Z+,≤) is a total order – a.k.a. totally ordered poset, linear order, or chain Discrete Structure/AM103 Dr. Manoj Gaur
• 9. 9 • In the poset (Z+,|) with “divides” operator |, are the integers 3 and 9 comparable? – Yes, as 3 | 9 • Are 7 and 5 comparable? – No, as 7 | 5 and 5 | 7 • Thus, as there are pairs of elements in Z+ that are not comparable, the poset (Z+,|) is a partial order. It is not a chain. Discrete Structure/AM103 Dr. Manoj Gaur
• 10. 10 (S, ) If is a poset and every two elements of S are comparable, S is called totally ordered or linearly ordered set, and is called a total order or a linear order. A totally ordered set is also called a chain. Totally Ordered, Chains Discrete Structure/AM103 Dr. Manoj Gaur
• 12. 12 Hasse Diagrams Given any partial order relation defined on a finite set, it is possible to draw the directed graph so that all of these properties are satisfied. This makes it possible to associate a somewhat simpler graph, called a Hasse diagram, with a partial order relation defined on a finite set. Discrete Structure/AM103 Dr. Manoj Gaur
• 13. 13 Hasse Diagram • For the poset ({1,2,3,4,6,8,12}, |) Discrete Structure/AM103 Dr. Manoj Gaur
• 14. 14 For the poset ({1,2,3,4,6,8,12}, |) Discrete Structure/AM103 Dr. Manoj Gaur
• 16. 16 a is a maximal in the poset (S, ) if there is no such that a b. Similarly, an element of a poset is called minimal if it is not greater than any element of the poset. That is, a is minimal if there is no element such that b a. It is possible to have multiple minimals and maximals. Maximal and Minimal Elements b S  b S  Discrete Structure/AM103 Dr. Manoj Gaur
• 21. 21 a is the greatest element in the poset (S, ) if b a for all . Similarly, an element of a poset is called the least element if it is less or equal than all other elements in the poset. That is, a is the least element if a b for all Greatest Element Least Element b S  b S  Discrete Structure/AM103 Dr. Manoj Gaur
• 22. 22 Upper bound, Lower bound Sometimes it is possible to find an element that is greater than all the elements in a subset A of a poset (S, ). If u is an element of S such that a u for all elements , then u is called an upper bound of A. Likewise, there may be an element less than all the elements in A. If l is an element of S such that l a for all elements , then l is called a lower bound of A. a A  a A  Discrete Structure/AM103 Dr. Manoj Gaur
• 23. 23 Upper Bound of: ub{2,6}={6,12} ub{4,8}={8} ub{2,6}= {6,12} Ub{3,4}={12} Ub{8,12}= no upper bound Ub{2,3,6}={6,12} Ub{1,2,4}={4,8,12} Lower Bounds: lb{2,6}={1,2} lb{4,8}={1,2,4} lb{3,4}={1} lb{8,12}= {4,2,1} lb{2,3,6}={1} lb{1,2,4}={1,2} Discrete Structure/AM103 Dr. Manoj Gaur
• 24. 24 Least Upper Bound, Greatest Lower Bound The element x is called the least upper bound (lub) of the subset A if x is an upper bound that is less than every other upper bound of A. The element y is called the greatest lower bound (glb) of A if y is a lower bound of A and z y whenever z is a lower bound of A. Discrete Structure/AM103 Dr. Manoj Gaur
• 25. 25 Upper Bound and least upper bound of: ub{2,6}={6,12} lub{2,6}={6} ub{4,8}={8} lub{4,8}={8} ub{2,6}= {6,12} Ub{3,4}={12} Ub{8,12}= no upper bound Ub{2,3,6}={6,12} lub{2,3,6}={6} Ub{1,2,4}={4,8,12} lub{1,2,4}={4} Lower Bounds and greatest lower bounds: lb{2,6}={1,2} glb{2,6}={2} lb{4,8}={1,2,4} glb{4,8}={4} lb{3,4}={1} glb{3,4}={1} lb{8,12}= {4,2,1} glb{8,12}={4} lb{2,3,6}={1} glb{2,3,6}={1} lb{1,2,4}={1,2} Discrete Structure/AM103 Dr. Manoj Gaur
• 28. 28 Lattices A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice. Discrete Structure/AM103 Dr. Manoj Gaur
• 30. 30 Examples 21 and 22, p. 575 in Rosen. Discrete Structure/AM103 Dr. Manoj Gaur
Aktuelle SpracheEnglish
Español
Portugues
Français
Deutsche