1) The document discusses partial orderings and partially ordered sets (posets). A poset is a set together with a partial ordering relation that is reflexive, antisymmetric, and transitive. 2) Examples of posets are provided, including the power set of {a,b,c} under subset relation. The concepts of comparable and incomparable elements in a poset are explained. 3) Additional concepts discussed include maximal and minimal elements, greatest and least elements, upper and lower bounds, and least upper bounds and greatest lower bounds.

1. DISCRETE STRUCTURES
AM103
Department of Applied Sciences
Chitkara University, Punjab

2. Lattice
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Discrete Structure/AM103 Dr. Manoj Gaur

3. Partial Orderings
• A partial ordering (or partial order) is a relation that is
reflexive, antisymmetric, and transitive
– Recall that antisymmetric 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
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Discrete Structure/AM103 Dr. Manoj Gaur

4. Partially Ordered Set (POSET)
• A relation R on a set S is called a partial ordering or partial order if
it is reflexive, antisymmetric, and transitive. A set S together with a
partial ordering R is called a partially ordered set, or poset, and is
denoted by (S, R)
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Discrete Structure/AM103 Dr. Manoj Gaur

5. Example (1)
• Let S = {1, 2, 3} and
• let R = {(1,1), (2,2), (3,3), (1, 2), (3,1), (3,2)}
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1
2
3
Discrete Structure/AM103 Dr. Manoj Gaur

6. Comparable / Incomparable
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Discrete Structure/AM103 Dr. Manoj Gaur

7. Example
• Consider the power set of {a, b, c} and the subset relation.
(P({a,b,c}), )
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{ , } { , } and { , } { , }
a c a b a b a c
 
So, {a,c} and {a,b} are incomparable

Discrete Structure/AM103 Dr. Manoj Gaur

8. 8
• 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, or chain
Discrete Structure/AM103 Dr. Manoj Gaur

9. 9
• In the poset (Z+,|) with “divides” operator |, 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. It is not a chain.
Discrete Structure/AM103 Dr. Manoj Gaur

10. 10
(S, )
If is a poset and every two elements of
S are comparable, S is called totally ordered or
linearly ordered set, and is called a total
order or a linear order. A totally ordered set is
also called a chain.
Totally Ordered, Chains
Discrete Structure/AM103 Dr. Manoj Gaur

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Discrete Structure/AM103 Dr. Manoj Gaur

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Hasse Diagrams
Given any partial order relation defined on a finite set,
it is possible to draw the directed graph so that all
of these properties are satisfied.
This makes it possible to associate a somewhat simpler
graph, called a Hasse diagram, with a partial order
relation defined on a finite set.
Discrete Structure/AM103 Dr. Manoj Gaur

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Hasse Diagram
• For the poset ({1,2,3,4,6,8,12}, |)
Discrete Structure/AM103 Dr. Manoj Gaur

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For the poset ({1,2,3,4,6,8,12}, |)
Discrete Structure/AM103 Dr. Manoj Gaur

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Discrete Structure/AM103 Dr. Manoj Gaur

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a is a maximal in the poset (S, ) if there is no
such that a b. Similarly, an element of a poset is
called minimal if it is not greater than any element of
the poset. That is, a is minimal if there is no element
such that b a.
It is possible to have
multiple minimals and maximals.
Maximal and Minimal Elements
b S

b S

Discrete Structure/AM103 Dr. Manoj Gaur

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Discrete Structure/AM103 Dr. Manoj Gaur

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Discrete Structure/AM103 Dr. Manoj Gaur

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Discrete Structure/AM103 Dr. Manoj Gaur

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Discrete Structure/AM103 Dr. Manoj Gaur

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a is the greatest element in the poset (S, ) if b a
for all . Similarly, an element of a poset is called
the least element if it is less or equal than all other
elements in the poset. That is, a is the least element if
a b for all
Greatest Element
Least Element
b S

b S

Discrete Structure/AM103 Dr. Manoj Gaur

22. 22
Upper bound, Lower bound
Sometimes it is possible to find an element that is
greater than all the elements in a subset A of a poset
(S, ). If u is an element of S such that a u for all
elements , then u is called an upper bound of A.
Likewise, there may be an element less than all the
elements in A. If l is an element of S such that l a
for all elements , then l is called a lower bound of
A.
a A

a A

Discrete Structure/AM103 Dr. Manoj Gaur

23. 23
Upper Bound of:
ub{2,6}={6,12}
ub{4,8}={8}
ub{2,6}= {6,12}
Ub{3,4}={12}
Ub{8,12}= no upper bound
Ub{2,3,6}={6,12}
Ub{1,2,4}={4,8,12}
Lower Bounds:
lb{2,6}={1,2}
lb{4,8}={1,2,4}
lb{3,4}={1}
lb{8,12}= {4,2,1}
lb{2,3,6}={1}
lb{1,2,4}={1,2}
Discrete Structure/AM103 Dr. Manoj Gaur

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Least Upper Bound,
Greatest Lower Bound
The element x is called the least upper bound (lub) of
the subset A if x is an upper bound that is less than
every other upper bound of A.
The element y is called the greatest lower bound (glb)
of A if y is a lower bound of A and z y whenever z is
a lower bound of A.
Discrete Structure/AM103 Dr. Manoj Gaur

25. 25
Upper Bound and least upper bound of:
ub{2,6}={6,12} lub{2,6}={6}
ub{4,8}={8} lub{4,8}={8}
ub{2,6}= {6,12}
Ub{3,4}={12}
Ub{8,12}= no upper bound
Ub{2,3,6}={6,12} lub{2,3,6}={6}
Ub{1,2,4}={4,8,12} lub{1,2,4}={4}
Lower Bounds and greatest lower
bounds:
lb{2,6}={1,2} glb{2,6}={2}
lb{4,8}={1,2,4} glb{4,8}={4}
lb{3,4}={1} glb{3,4}={1}
lb{8,12}= {4,2,1} glb{8,12}={4}
lb{2,3,6}={1} glb{2,3,6}={1}
lb{1,2,4}={1,2}
Discrete Structure/AM103 Dr. Manoj Gaur

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Discrete Structure/AM103 Dr. Manoj Gaur

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Discrete Structure/AM103 Dr. Manoj Gaur

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Lattices
A partially ordered set in which every pair
of elements has both a least upper bound
and a greatest lower bound is called a
lattice.
Discrete Structure/AM103 Dr. Manoj Gaur

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Lattices
Discrete Structure/AM103 Dr. Manoj Gaur

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Examples 21 and 22, p. 575 in Rosen.
Discrete Structure/AM103 Dr. Manoj Gaur

31. 31
Z+={1,2,3…………..}
Discrete Structure/AM103 Dr. Manoj Gaur