8.3 Representing Relations

• Directed Graphs– Vertex– Arc (directed edge)– Initial vertex– Terminal vertex

Example

• Draw the “divides” relation on the set {2,3,4,5,6,7,8,9} as a directed graph

The zero-one Matrix Representation MR

• MR is just a zero-one version of the “chart” representation of R.

2 3 4 5 6 7 8 92 1 0 1 0 1 0 1 03 0 1 0 0 1 0 0 14 0 0 1 0 0 0 1 05 0 0 0 1 0 0 0 06 0 0 0 0 1 0 0 07 0 0 0 0 0 1 0 08 0 0 0 0 0 0 1 09 0 0 0 0 0 0 0 1

Reflexivity

Directed graph picture Zero-one matrix picture

Symmetry

Directed graph picture Zero-one matrix picture

Antisymmetry

Directed graph picture Zero-one matrix picture

Transitivity

Directed graph picture Zero-one matrix picture

Theorem

1221

2121

2121

and ,

,

RRRR

RRRR

RRRR

MMM

MMM

MMM

For relations R1 and R2 on set A,

Example: Let and be binary relations. Find and . Use them to find Verify by calculating without matrices.

Corollary

][nRR

MM n

For a relation R on set A,

for any positive integer n.

Example: Let and . Calculate and to determine if the relations and are transitive.

8.4 Closures of Relations

• Reflexive closure

• Symmetric closure

Example: Let R be the relation on the set containing the pairs

What is the reflexive closure of R? What is the symmetric closure of R?

{ (1,1 ) , (1,2 ) , (2,4 ) , (3,1 ) , (4,2 ) }

Paths in Directed Graphs

• A path in a directed graph is a sequence of vertices for which any two consecutive vertices ai and ai+1 in the sequence are joined by an arc from ai to ai+1.

• Theorem: Let R be a relation on set A, and n a positive integer. Then there is a path of length n from a to b in R if and only if (a,b) is in Rn.

Example:

The “Connectivity Relation” R*

• Let R be a relation on set A. We define

1

*

i

iRR

Example: Let be the relation on the set of all people in the world that contains if has met . What is , where is a positive integer greater than one? What is ?

The Transitive Closure

• For a relation R on a set A, we define the transitive closure of R to be the smallest transitive relation containing R.

• Theorem:

Finding transitive closure the “hard” way: 𝑅={ (1,1 ) , (1,2 ) , (2,4 ) , (3,1 ) , (4,2 ) }

Computing R*

• If A is a set with n elements, and R is a relation on A, then any time there is a path of length one or more from a to b in R then there is a path of length n or less.

• So actually

and

• Interestingly, this is not the best way of computing R*

n

i

iRR1

*

][

1*

iR

n

iRMM

𝑅={ (1,1 ) , (1,2 ) , (2,4 ) , (3,1 ) , (4,2 ) }Computing transitive closure the better way:

𝑀𝑅=[1 1 00 0 010

01

00

0100] 𝑀𝑅❑

[2]=[ 1 1 00 1 010

10

00

1001]

𝑀𝑅❑[3 ]=[1 1 0

0 0 010

11

00

1110] 𝑀𝑅❑

[4 ]=[1 1 00 1 010

10

00

1011]

⋁

⋁ ⋁

¿𝑀𝑅∗=[1 1 00 1 010

11

00

1111]

Warshall’s Algorithm

procedure Warshall(MR: n by n zero-one matrix)

W := MR

for k:=1 to n for i:=1 to n for j:=1 to n wij := wij (wik wkj)

{ W now contains MR* }

Illustration of Warshall’s Algorithm

𝑀𝑅=[1 1 00 0 010

01

00

0100] [1 1 0

0 0 010

01

00

0100][1 1 0

0 0 010

01

00

1100] [1 1 0

0 0 010

01

00

1100][1 1 0

0 1 010

01

00

1100]

[1 1 00 1 010

01

00

1100][1 1 0

0 1 010

11

00

1100] [1 1 0

0 1 010

11

00

1100][1 1 0

0 1 010

11

00

1110] [1 1 0

0 1 010

11

00

1110][1 1 0

0 1 010

11

00

1111][1 1 0

0 1 010

11

00

1111]

8.5 Equivalence Relations

• Definition: A relation R on set A is an equivalence relation if …

Examples

• aRb if and only if a and b have the same first name (on the set of students in this class)

• aRb if and only if a ≡ b (mod 5) (on the set of integers)

Equivalence Classes

• If R is a relation on set A, and a is an element of A, then…

Examples (continued)

• [Michael]

• [4]5

Theorem

For an equivalence relation R on set A and elements a and b of A, the following are all logically equivalent:

a) a R b

b) [a]R = [b]R

c) [a]R [b]R

Partitions

• For a set S, a partition of S is a collection = {A1, A2, …, Am} of nonempty subsets of S which satisfies the following properties:

– Every element of A is in one of the sets Ai.– For all i, j {1, 2, …, m},

if i j then Ai Aj = • Terminology: We say that the collection

partitions S.

Theorem

• Let R be an equivalence relation on set S. Then the equivalence classes of R partition S. Conversely, for any partition of S there is an equivalence relation R whose equivalence classes are the sets in .

…, , ,-8, , ,-5, , ,-2, , ,1, , ,4, , ,7, , ,10, ……, ,-9, , ,-6, , ,-3, , ,0, , ,3, , ,6, , ,9, , ……,-10, , ,-7, , ,-4, , ,-1, , ,2, , ,5, , ,8, , , ……,-10, , ,-7, , ,-4, , ,-1, , ,2, , ,5, , ,8, , , ……, ,-9, , ,-6, , ,-3, , ,0, , ,3, , ,6, , ,9, , ……, , ,-8, , ,-5, , ,-2, , ,1, , ,4, , ,7, , ,10, …

Visual

[2]={… ,-10, -7, -4, -1, 2, 5, 8, …}

[0]={… ,-9, -6, -3, 0, 3, 6, 9, …}

[1]={… ,-8, -5, -2, 1, 4, 7, 10, …}

…, , ,-8, , ,-5, , ,-2, , ,1, , ,4, , ,7, , ,10, …

…, ,-9, , ,-6, , ,-3, , ,0, , ,3, , ,6, , ,9, , …

Visual

…,-10, , ,-7, , ,-4, , ,-1, , ,2, , ,5, , ,8, , , …

[2]={… ,-10, -7, -4, -1, 2, 5, 8, …}

[0]={… ,-9, -6, -3, 0, 3, 6, 9, …}

[1]={… ,-8, -5, -2, 1, 4, 7, 10, …}

Example: Turning a partition into an equivalence relation

𝐴1= {1,5 } ,𝐴2={2,3,6 } ,𝐴3={ 4 }𝑜𝑛 h𝑡 𝑒𝑠𝑒𝑡 𝑆={1,2,3,4,5,6 }

Zero-One Matrix Representation of an Equivalence Relation Examples

Digraph Representation of an Equivalence Relation Examples

Equivalence as “sameness”

• Almost every equivalence relation definition comes down to identifying some notion of “sameness”– Same remainder when divided by n– Same name– Same set of a partition

Number of Partitions of a Set with n Elements

• n = 1

• n = 2

• n = 3

• n = 4

Recurrence Relation for the Number of Partitions of a Set with n Elements

8.6 Partial Orderings

Let A be a set, and R a relation on A. We say that R is a partial ordering if and only if…

In this case we say that the pair (A, R) is a partially ordered set (poset).

Examples:

1. The real numbers R under the relation2. The real numbers R under the relation3. The positive integers under the “divides”

relation4. Any set of sets under the (subset)

relation5. The cartesian product ZZ under the

“(,)” relation R. (i.e. (x,y) R (z,w) if and only if x z and y w.)

Convention

• The symbol is the default symbol used to represent a partial ordering.

• Example: “Let A be a set, and let be a partial ordering on A.”

Comparable and Incomparable Elements

• Two elements a and b of a partially ordered set are said to be incomparable if and only if the statements a b and b a are both false. Otherwise the elements are comparable.

• Examples: – Subsets– Cartesian products– Divides relation

Examples:

Total Orderings

• Let A be a set, and let be a partial ordering on A. We say that is a total ordering provided…

In this case we say that the pair (A, ) is a totally ordered set. (linearly ordered set, chain)

• Examples: 1) Real numbers under 2) Any set of strings under the “dictionary”, or lexicographic, ordering

A partial order on “induced” by partial orders on and on

• Lexicographic ordering

• Example:

Well-Ordered Sets

• A set S is well-ordered by the partial ordering if and only if every nonempty subset of S has a least element (minimum element).

• Examples:

• Non-Examples: , ,

Hasse Diagrams

• Begin with the digraph representation of the partial ordering

• Omit the reflexive loops• Omit all edges which would be implied by

transitivity• Orient all vertices and arcs so that the

direction of each arc is up.• Remove the direction arrow from each arc

Hasse Diagram Example I

• Pairs in {1,2,3}{1,2,3} under lexicographic order

Hasse Diagram Example I

• Pairs in {1,2,3}{1,2,3} under lexicographic order

Hasse Diagram Example I

• Pairs in {1,2,3}{1,2,3} under lexicographic order

• Pairs in {1,2,3}{1,2,3} under lexicographic order

Hasse Diagram Example I

• Pairs in {1,2,3}{1,2,3} under lexicographic order

Hasse Diagram Example I

Hasse Diagram Example II

• Integers 1-12 under “divides”

Hasse Diagram Example III

• Integers 1-12 under “divides”

Maximal and Minimal ElementsGiven a poset and an element , we say

• is maximal in the poset if

• is minimal in the poset if

• is the greatest element of the poset if

• is the least element of the poset if

Theorems

• Every finite poset has a minimal element

• Every finite poset has a maximal element

Example:

Maximal Elements?

Minimal Elements?

Greatest Element?

Least Element?

Upper Bounds and Lower Bounds

Given a subset of a poset we say

• is an upper bound of if

• is a lower bound of if

• is the least upper bound of if

• is the greatest lower bound of if

Example:Find the upper and lower bounds of the subsets {c,f}, {h,i}, {c,d,e}.

Find the greatest lower bound and the least upper bound of {b,d} and {a,b,g}.

Topological Sorts

• Let (A,) be a partial ordering, and let be a total ordering on the same set A. We say that is a compatible total ordering for provided that for all a, b in A, whenever a b then also a b.

• A topological sort is an algorithm which, given a partially ordered set, generates a compatible total ordering. In other words, it generates the elements of A one by one in a linear order compatible with the partial ordering. (PERT charts)

Topological Sort Algorithm

Procedure TopologicalSort(S: finite poset)

Let q be a queue of elements of S, initially empty

While S is not empty, do begin

Choose and enqueue a minimal element s of

S onto the queue q.

Remove s from S

End

{ The queue q now contains all the elements of S, arranged in a compatible total ordering of S. }

Topological Sort Trace