Poset in Relations(Discrete Mathematics)

1. Relations - POSET
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 explain POSET in Relations

3. Prerequisite
• Basics of Discrete Mathematics
• Basics of Relation (for reference go to video 1)
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4. Introduction
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• Partial Ordered Set (POSET) :
• A relation R on a set A is called a partial order if R is reflexive,
anti-symmetric and transitive.
• The set A together with the partial order R is called a partially
ordered set, or simply a poset, denoted by (A, R)
• Let A be a collection of subsets of a set S. The relation ⊆ of set
inclusion is a partial order on A, so (A, ⊆ ) is a poset.

5. Example
• Show that the relation “greater than or equal to” is a partial ordering
on the set of integers.
Solution :
Let Z be the set of all integers and the relation R = ‘≥’
• Since a ≥ a for every integer a, the relation ‘≥’ is reflexive.
• Let a and b be any two integers.
• Let aRb and bRa => a ≥ b and b ≥a
• => a=b
∴ The relation ‘≥’ is antisymmetric.
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6. Let a,b and c be any three integers.
Let a ≥ b and b ≥ c
a ≥c
The relation ‘≥’ is transitive.
Since, the relation is transitive, reflexive and anti-symmetric, ≥ is
a partial ordering on the set of integers.
Therefore,(Z, ≥) is a POSET.
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7. Walchand Institute of Technology, Solapur
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Partial Ordering Relation
Transitive
Anti-
symmetric
Reflexive

8. Think & Write
Question : Can we say if any one or two conditions satisifies then
it is partial order?
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9. Answer
No, it should satisfy all the three conditions.
Transitive
Reflexive
Anti-symmetric
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10. Example
For,
Set A ={1,2,3}
R1 = {}
R2 = { (1,1)(2,2)(3,3)}
R3 = {(1,1)(2,2)(3,3)(1,2)(2,1)}
R4 = {(1,1)(2,2)(3,3)(1,3)(2,3)}
R5 = {(1,1)(1,2)(2,3)(1,3)}
R6 = {(1,1)(1,3)(2,2)(2,3)(3,3)}
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11. Conclusion :
In this session, We have studied all about POSET.
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12. 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. SNTHA
(CENGAGE Learning)
• 2. Discrete Mathematical Structures – Bernard Kolman, Robert C. Busby (Pearson
Education)
• 3. Discrete mathematics -- Liu (MGH)
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13. Thank You !!!
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