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DISCRETE STRUCTURES
AM103
Department of Applied Sciences
Chitkara University, Punjab
Lattice
2
Discrete Structure/AM103 Dr. Manoj Gaur
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
3
Discrete Structure/AM103 Dr. Manoj Gaur
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)
4
Discrete Structure/AM103 Dr. Manoj Gaur
Example (1)
• Let S = {1, 2, 3} and
• let R = {(1,1), (2,2), (3,3), (1, 2), (3,1), (3,2)}
5
1
2
3
Discrete Structure/AM103 Dr. Manoj Gaur
Comparable / Incomparable
6
Discrete Structure/AM103 Dr. Manoj Gaur
Example
• Consider the power set of {a, b, c} and the subset relation.
(P({a,b,c}), )
7
{ , } { , } and { , } { , }
a c a b a b a c
 
So, {a,c} and {a,b} are incomparable

Discrete Structure/AM103 Dr. Manoj Gaur
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
• 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
(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
11
Discrete Structure/AM103 Dr. Manoj Gaur
12
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
13
Hasse Diagram
• For the poset ({1,2,3,4,6,8,12}, |)
Discrete Structure/AM103 Dr. Manoj Gaur
14
For the poset ({1,2,3,4,6,8,12}, |)
Discrete Structure/AM103 Dr. Manoj Gaur
15
Discrete Structure/AM103 Dr. Manoj Gaur
16
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
17
Discrete Structure/AM103 Dr. Manoj Gaur
18
Discrete Structure/AM103 Dr. Manoj Gaur
19
Discrete Structure/AM103 Dr. Manoj Gaur
20
Discrete Structure/AM103 Dr. Manoj Gaur
21
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
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
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
24
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
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
26
Discrete Structure/AM103 Dr. Manoj Gaur
27
Discrete Structure/AM103 Dr. Manoj Gaur
28
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
29
Lattices
Discrete Structure/AM103 Dr. Manoj Gaur
30
Examples 21 and 22, p. 575 in Rosen.
Discrete Structure/AM103 Dr. Manoj Gaur
31
Z+={1,2,3…………..}
Discrete Structure/AM103 Dr. Manoj Gaur

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AM103(Unit3).pptx

  • 1. DISCRETE STRUCTURES AM103 Department of Applied Sciences Chitkara University, Punjab
  • 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 3 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) 4 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)} 5 1 2 3 Discrete Structure/AM103 Dr. Manoj Gaur
  • 6. Comparable / Incomparable 6 Discrete Structure/AM103 Dr. Manoj Gaur
  • 7. Example • Consider the power set of {a, b, c} and the subset relation. (P({a,b,c}), ) 7 { , } { , } 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
  • 12. 12 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
  • 13. 13 Hasse Diagram • For the poset ({1,2,3,4,6,8,12}, |) Discrete Structure/AM103 Dr. Manoj Gaur
  • 14. 14 For the poset ({1,2,3,4,6,8,12}, |) Discrete Structure/AM103 Dr. Manoj Gaur
  • 16. 16 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
  • 21. 21 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
  • 24. 24 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
  • 28. 28 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
  • 30. 30 Examples 21 and 22, p. 575 in Rosen. Discrete Structure/AM103 Dr. Manoj Gaur