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Binary Relations: Properties, Operations, and Diagrams, Study notes of Discrete Structures and Graph Theory

University of MarylandDiscrete Structures and Graph Theory

The fundamental concepts of relations, their properties, inverse, complement, closures, matrix representation, union, intersection, difference, and composition. Topics include binary relations, reflexivity, symmetry, transitivity, equivalence relations, and partial order relations. The document also introduces hasse diagrams for visualizing partial order sets.

Typology: Study notes

Pre 2010

Uploaded on 02/13/2009

koofers-user-g1p
koofers-user-g1p 🇺🇸

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Relations
• binary relations xRy
• on sets x∈X y∈Y R ⊆XY
• Example:
“less than” relation from A={0,1,2} to B={1,2,3}
use traditional notation
0 < 1, 0 < 2, 0 < 3, 1 < 2, 1 < 3, 2 < 3
1 1, 2 1, 2 2
or use set notation
AB={(0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)}
R={(0,1),(0,2),(0,3), (1,2),(1,3), (2,3)}
or use Arrow Diagrams
Formal Definition
• (binary) relation from A to B
where x∈A, y∈B, (x,y)∈AB and R⊆AB
xRy ↔(x,y) ∈R
• finite example: A={1,2} B={1,2,3}
• infinite example: A = Z and B = Z
aRb ↔a-b∈Zeven
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Relations

  • binary relations xRy
  • on sets x∈X y∈Y R ⊆ XY
  • Example: “less than” relation from A={0,1,2} to B={1,2,3} use traditional notation 0 < 1, 0 < 2, 0 < 3, 1 < 2, 1 < 3, 2 < 3 1  1, 2  1, 2  2 or use set notation AB={(0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)} R={(0,1),(0,2),(0,3), (1,2),(1,3), (2,3)} or use Arrow Diagrams

Formal Definition

  • (binary) relation from A to B where x∈A, y∈B, (x,y)∈AB and R⊆ AB xRy ↔ (x,y) ∈R
  • finite example: A={1,2} B={1,2,3}
  • infinite example: A = Z and B = Z aRb ↔ a-b∈Zeven

Properties of Relations

  • Reflexive
  • Symmetric
  • Transitive

R isReflexive↔ ∀ x ∈ A, xRx

R isSymmetric↔ ∀ x , y ∈ A, xRy → yRx

R isTransitive↔ ∀ x , y , z ∈ A, xRy ∧ yRz → xRz

Proving Properties

on Infinite Sets

  • m,n ∈Z, m ≡ 3 n
  • Reflexive
  • Symmetric
  • Transitive

Matrix Representation

of a Relation

  • MR = [mij] mij={1 iff (i,j) ∈R and 0 iff (i,j)∉R
  • example: R : {1,2,3} →{1,2} R = {(2,1),(3,1),(3,2)}

=^ 

M R

Powers of Relation

  • For relation M,
    • M^1 = the list of paths available in 1 step
    • M^2 = the list of paths available in 2 steps
    • M^3 = the list of paths available in 3 steps
    • …
    • M*^ = the list of paths available in any number of steps
  • through composition
  • through matrix multiplication

Union, Intersection, Difference

and Composition

  • R: A→B and S: A→B
  • R: A→B and S: B→C

R  S ={( x , y )∈ A × B |( x , y )∈ R ∨( x , y )∈ S }

R  S ={( x , y )∈ A × B |( x , y )∈ R ∧( x , y )∈ S }

R − S ={( x , y )∈ A × B |( x , y )∈ R ∧( x , y )∉ S }

S − R ={( x , y )∈ A × B |( x , y )∈ S ∧( x , y )∉ R }

S  R ={( a , c )∈ A × C |∃ b ∈ B ,( a , b )∈ R ∧( b , c )∈ S }

Equivalence Relations

  • Partition the elements: any elements “related” are in the same partition
  • Equivalence Relations are
    • Reflexive
    • Symmetric
    • Transitive
  • Partitions are called Equivalence Classes
    • [a] = equivalence class containing a
    • [a] = {x ∈A | xRa}

Partial Order Relation

  • R is a Partial Order Relation if and only if R is Reflexive, Antisymmetric and Transitive
  • Partial Order Set (POSET) (S,R) = R is a partial order relation on set S
  • Examples
    • (Z,)
    • (Z+,|) {note: | symbolizes divides}

Total Ordering

  • When all pairs from the set are “comparable” it is called a Total Ordering
  • a and b are comparable if and only if a R b or b R a
  • a and b are non-comparable if and only if a R b and b R a

Hasse Diagram

  1. take the digraph (since it represents the same relation)
  2. arrange verticies so all arrows go upward (since it is antisymmetric we know this is possible)
  3. remove the reflexive loops (since we know it is reflexive these are not necessary)
  4. remove the transitive arrows (since we know it is transitive, these are not necessary)
  5. make the remaining edges non-directed (since we know they are all going upward, the direction is not necessary)

Hasse Diagram Example

  • The POSET ({1,2,3,9,18},|)
  • The POSET ({1,2,3,4},)
  • The POSET (P{a,b,c},⊆)
  • Draw complete digraph diagrams of these relations
  • Derive Hasse Diagram from those