The document is a presentation on discrete mathematics that discusses partially ordered sets (posets). It introduces posets and defines the properties of a partial order relation. It explains how posets can be represented diagrammatically using Hasse diagrams and defines the key elements in a poset including least/greatest elements, minimal/maximal elements, and bounds. It also describes linearly ordered sets where every pair of elements is comparable and well ordered sets which are totally ordered and every non-empty subset has a least element.

1. AMRAPALI INSTITUTE OF TECHNOLOGY AND
SCIENCES
A PRESENTATION ON DISCRETE
MATHEMATICS
ON
“ POSETS”
PRESENTED BY:
Richa dasila

2. CONTENTS
 Introduction.
 Diagrammatic Representation.
 Elements in Posets.
 Linear Ordered Posets and Well Ordered Posets.

3. INTRODUCTION.
A relation “R” on set “P” is a partial order relation if
P is:
Reflexive .
Anti-symmetric .
Transitive .
• The set P with partial order relation (P,<)is called a
Poset.

4. DIAGRAMMATIC
REPRESENTATION.
 Hasse Diagram is used to represent poset .
e.g.,
A = {2,3,4,5,6,8,24} and the relation is divides then ,
(A,|)= {(2,4),(2,6),(3,6),(4,8),(6,24),(8,24)} is covering relation .
Hasse Diagram :
36
12 18
6
2 3

5. ELEMENTS IN POSET .
Elements
In
POSET
Least Element .-
if “a” precedes x
for all x £ A .
Greatest
Element : if “a”
succeeds x for
all x£ A .
Minimal
Element : “a”
exists such that
no element
precedes “a” .
Maximal
Element : “a”
exists such that
no element x
succeeds “a”.

6. BOUNDS IN POSET .
Lower and Upper Bounds :
“a” is lower bound if a
precedes “x ” for all x in
A(set) ; “a” is upper bound
if it succeeds “x” for all x in
A.
GLB and LUB : If set of
lower bounds of A have a
greatest element then it is
GLB and if the set of upper
bounds of A have a least
element then it is LUB .

7. LINEARLY ORDERED SET
AND WELL – ORDERED SET
.
 Linearly Ordered Set : If every pair of elements in poset P is
comparable .
 Well Ordered Set : If the partial order relation is totally ordered
and every non – empty subset of P has a least element .