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Understanding Partially Ordered Sets (Poset) and Hasse Diagrams

Explore the concept and applications of partially ordered sets (Poset) in daily life, with a focus on creating orders and understanding Hasse Diagrams. Learn about the basics of sets, criteria, terms used, and topological sorting in Poset. Discover how Poset is utilized in comparison matters like a dictionary and represent relationships graphically through Hasse Diagrams.

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Understanding Partially Ordered Sets (Poset) and Hasse Diagrams

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  1. CSNB 143 Discrete Mathematical Structures Chapter 9 – Poset

  2. POSET • OBJECTIVES • Student should be able to understand the concept used in dictionary. • Students should be able to apply poset in daily lives that involves order. • Students should be able to create order by themselves.

  3. What, Which, Where, When • Basics of set • Its criteria (Clear / Not Clear) • Terms used in poset (Clear / Not Clear) • Hasse Diagram (Clear / Not Clear) • Topological Sorting (Clear / Not Clear)

  4. PARTIALLY ORDERED SETS (POSET) • A relation R on set A is called Partial Order if R is reflexive, antisymmetric and transitive. • In short, it is called Poset, written as (A, R) where R is a relation that turns A to a poset. • Poset is being used widely in comparison matters such as a dictionary. • Ex 1: Let say set S = {a, b, c,… z} is an ordered set. Then set S* is a set for all possibility of words in various length, either it is meaningful or not. So we can get

  5. help < helping • In S* because help < help in s5. • And also helper < helping • because helpe < helpi in s5. • Using this comparison, a dictionary was introduced. • Theorem: A diagraph for poset has no cycle length more that 1.

  6. Hasse Diagram • Let A is a finite set. • To draw a diagraph for poset, we must consider three things: • Graph has no cycle length 1 (irreflexive). • Graph is not transitive for all vertices. • Graf has no arrow (always pointing upwards) • This particular graph is called a Hasse Diagram. • Hasse Diagram is one of the methods to represent poset.

  7. Ex2: Consider a diagraph below: • Then, consider the things to make it a Hasse Diagram. B C A

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