The document provides an overview of partially ordered sets (posets) and their properties, including the definitions of reflexivity, antisymmetry, and transitivity necessary for a relation to be classified as a partial order. It illustrates various examples of posets, such as integers under different relations and the inclusion relation on power sets, and discusses the concept of comparability among elements. Additionally, it explains the distinction between partially ordered sets and totally ordered sets, highlighting that not all posets have comparable elements.

1. DISCRETE STRUCTURES AND THEORY OF LOGIC (DSTL)
KCS-303
Unit-3
Introduction to Poset
1

2. Partially Ordered Set (Poset)
2
Partial Ordering: A relation R on a set A is said to be partial
ordering or partial order if R is reflexive, antisymmetric and
transitive. i.e.
i. aRa for all a  A
ii. aRb and bRa  a = b
iii. aRb and bRc  aRc
Poset: A set A together with a partial order relation R is called a
partially ordered set or Poset. It is denoted by (A, R).
For the relation R, we use a relation “ ≤ “ called the usual order.

3. Example
3
Example: The relation ≥ is a partial ordering on the set of
integers Z and (Z, ≥) is a poset.
Example: The set of all positive integers Z+
under divisibility
forms a poset. i.e. (Z+
,  ) is a poset.
Example: Show that the inclusion relation  is a partial
ordering on the power set of a set S.
Sol. We can see that
a) A R A  A  A, whenever A is a subset of S, which is true.
  is reflexive

4. Example
4
b) A R B and B R A  A  B and B  A  A = B
  is antisymmetric
c) A R B and B R C  A  B and B  C  A  C  A R C
  is transitive
  is a partial ordering on P(S).
Hence (P(S),  ) is a poset.
Q. Let R is a relation defined as R = {(a, b)|a  b} on the set of
integers. Is R partial order relation?
Sol. Yes

5. Example
5
Q. Let R is a relation defined as R = {(a, b)|a < b} on the set of
integers. Is R partial order relation?
Sol. No.
We can see that
For any integer a, It is not true that a < a.
(a, a)  R. which shows R is not reflexive.
Hence R is not partial order relation.

6. Comparability
6
Comparability: The elements a and b of a poset (A, ≤ )are called
comparable if either a ≤ b or b ≤ a.
if neither a ≤ b nor b ≤ a, then a and b are called non
comparable.
Example: the relation of divisibility i.e. a R b if a  b, is a partial
ordering on Z+. The integers 2 and 4 are comparable while 2
and 5 are non comparable.

7. Totally Ordered Set
7
Totally Ordered (Linearly Ordered) Set: An Ordered set A is said
to be totally ordered if every pair of elements of A are
comparable.
Here A is also called a chain.
Example: The poset (Z+
,  ) is totally ordered.
Example: The poset (Z,  ) is totally ordered.
Example: The poset (Z+
,  ) is not totally ordered as it contains
elements that are non comparable.

8. Thank You
8