The document explains the concepts of equivalence relations and partial order relations, highlighting their properties like reflexivity and transitivity. It discusses examples of partial orders, such as the 'greater than or equal to' relation on integers and the divisibility relation on positive integers, as well as the concept of comparability within partially ordered sets (posets). Additionally, it introduces well-ordered sets and Hasse diagrams as visual representations of finite posets.

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M E G H A J K U M A R M A L L I C K
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PARTIAL ORDER RELATION

2. INTRODUCTION
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 An equivalence relation is a relation that is reflexive,
symmetric, and transitive
 A partial ordering (or partial order) is a relation that is
reflexive, anti -symmetric, and transitive
 Recall that anti-symmetric 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
 A set S with a partial ordering R is called a partially
ordered set, or poset
 Denoted by (S,R)

3. PARTIAL ORDERING EXAMPLES
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 Show that ≥ is a partial order on the set of integers
 It is reflexive: a ≥ a for all a  Z
 It is antisymmetric: if a ≥ b then the only way that
b ≥ a is when b = a
 It is transitive: if a ≥ b and b ≥ c, then a ≥ c
 Note that ≥ is the partial ordering on the set of
integers
 (Z, ≥) is the partially ordered set, or poset

4. SYMBOL USAGE
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 The symbol  is used to represent any relation
when discussing partial orders
 Not just the less than or equals to relation
 Can represent ≤, ≥,, etc
 Thus, a  b denotes that (a,b)  R
 The poset is (S,)
 The symbol  is used to denote a  b but a ≠ b
 If  represents ≥, then  represents >
 Fonts for this lecture set (specifically for the  and  symbols) are
available on the course website

5. COMPARABILITY
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 The elements a and b of a poset (S,) are called
comparable if either a  b or b  a.
 Meaning if (a,b)  R or (b,a)  R
 It can’t be both because  is antisymmetric
 Unless a = b, of course
 If neither a  b nor b  a, then a and b are incomparable
 Meaning they are not related to each other
 This is definition 2 in the text
 If all elements in S are comparable, the relation is a
total ordering

6. COMPARABILITY EXAMPLES
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 Let  be the “divides” operator |
 In the poset (Z+,|), 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

7. COMPARABILITY EXAMPLES
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 Let  be the less than or equals operator ≤
 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, chain, etc.

8. WELL-ORDERED SETS
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 (S,) is a well-ordered set if:
 (S,) is a totally ordered poset
 Every non-empty subset of S has at least element
 Example: (Z,≤)
 Is a total ordered poset (every element is comparable to every other
element)
 It has no least element
 Thus, it is not a well-ordered set
 Example: (S,≤) where S = { 1, 2, 3, 4, 5 }
 Is a total ordered poset (every element is comparable to every other
element)
 Has a least element (1)
 Thus, it is a well-ordered set

9. HASSE DIAGRAMS
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 Consider the graph for a finite poset ({1,2,3,4},≤)
 When we KNOW it’s a poset, we can simplify the
graph
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3
2
1
4
3
2
1
4
3
2
1
4
3
2
1
Called the
Hasse
diagram

10. HASSE DIAGRAM
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 For the poset ({1,2,3,4,6,8,12}, |)

11.  I T I S U S E D T O C A L C U L A T E T H E R A N K I N G O F
C O L L E G E S A C R O S S T H E W O R L D .
I T I S A L S O U S E D T O F I N D S C H E D U L E F O R V A R I O U S
S P O R T S .
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REAL LIFE EXAMPLE

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