This document discusses Hasse diagrams. It begins by stating the learning outcome is for students to illustrate Hasse diagrams. It then provides background on Hasse diagrams, noting they were originally devised to represent partially ordered sets and were created by Helmut Hasse. The document gives examples of drawing Hasse diagrams representing dividing relationships between numbers. It notes there can be multiple ways to draw a Hasse diagram for a given problem. In conclusion, it restates the topic covered was Hasse diagrams.

1. Hasse Diagrams
Ms. Rachana Pathak
(rachanarpathak@gmail.com)
Assistant Professor, Dept of Computer Science and Engineering
Walchand Institute of Technology, Solapur
(www.witsolapur.org)

2. Learning Outcome
2Walchand Institute of Technology, Solapur
At the end of this session,
Students will be able to illustrate the Hasse Diagrams.

3. Prerequisite
• Basics of Discrete Mathematics
• Basics of Relation
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4. History
• Hasse diagrams are named after Helmut Hasse (1898–1979)
• They are so called because of the effective use Hasse made of them.
• Hasse diagrams were originally devised as a technique for making drawings of
partially ordered sets (POSETS) by hand, they have more recently been created
automatically using graph drawing techniques.
• The phrase "Hasse diagram" may also refer to the transitive reduction as an abstract
directed acyclic graph, independently of any drawing of that graph.
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5. Introduction
A "GOOD" HASSE DIAGRAM -
• Although Hasse diagrams are simple as well as intuitive tools for
dealing with finite posets, it turns out to be rather difficult to draw
"good" diagrams.
• The reason is that there will be many possible ways to draw a Hasse
diagram for a given poset.
• The simple technique of just starting with the minimal elements of
an order and then drawing greater elements incrementally often
produces quite poor results: symmetries and internal structure of the
order are easily lost.
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6. Walchand Institute of Technology, Solapur 6

7. Walchand Institute of Technology, Solapur 7
Diagrammatical Representation :
• Hasse diagram is a type of mathematical diagram used to represent a finite
partially ordered set, in the form of a drawing of its transitive reduction.
• A partially ordered set (S, ≤) represents each element of S as a vertex in the
plane
•It draws a line segment or curve that goes upward from x to y whenever y
covers x (that is, whenever x < y and there is no z such that x < z < y).
• Curves may cross each other but must not touch any vertices other than their
endpoints.

8. Example#1
Draw the Hasse diagram representing the partial ordering
{(a,b)| a divides b} on {1,2,4,6,8,12}
Solution : Let P = {1,2,4,6,8,12}
(P, ≤ ) =
{(1,2),(1,3),(1,4),(1,6),(1,8),(1,12),(2,4),(2,6),(2,8),(2,12),(3,6),
(3,12),(4,8),(4,12),(6,12)}
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9. . .
. .
. .
.
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Example#1
1
2 3
4 6
8 12
ImageSource :1. Discrete Mathematics with combinatorics and graph theory- S. SANTHA (CENGAGE Learning)

10. Example#2
Let X = {2,3,6,12,24,36} and a relation ‘≤’ be such that x ≤ y if x
divides y. Draw the Hasse diagram of (x,≤).
Solution :
Given , X = {2,3,6,12,24,36} , ‘≤’ is a relation. The Hasse diagram
of (X,≤) is as shown.
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11. . .
.
.
. .
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2
3
6
12
24 36
ImageSource :1. Discrete Mathematics with combinatorics and graph theory- S. SANTHA (CENGAGE Learning)

12. Think & Write?
Can we draw more than one Hasse diagram for a given problem?
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13. Answer
Yes, we can draw Multiple hasse diagrams for single problem.
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14. Conclusion :
In this session, We have studied all about Hasse Diagrams.
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15. References
• 1. Discrete mathematical structures with applications to computer science -- J. P.
Tremblay & R. Manohar (MGH International)
• Reference Books:
• 1. Discrete Mathematics with combinatorics and graph theory- S. SANTHA
(CENGAGE Learning)
• 2. Discrete Mathematical Structures – Bernard Kolman, Robert C. Busby (Pearson
Education)
• 3. Discrete mathematics -- Liu (MGH)
15Walchand Institute of Technology, Solapur

16. Thank You !!!
16Walchand Institute of Technology, Solapur