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 M R M R is just a zero-one version of the “chart” representation of R

Reflexivity Directed graph pictureZero-one matrix picture

Symmetry Directed graph pictureZero-one matrix picture

Antisymmetry Directed graph pictureZero-one matrix picture

Transitivity Directed graph pictureZero-one matrix picture

Theorem For relations R 1 and R 2 on set A,

Corollary For a relation R on set A, for any positive integer n.

8.4 Closures of Relations Reflexive closure Symmetric closure

Paths in Directed Graphs A path in a directed graph is a sequence of vertices for which any two consecutive vertices a i and a i+1 in the sequence are joined by an arc from a i to a i+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 R n.

Example:

The “Connectivity Relation” R* Let R be a relation on set A. We define

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:

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*

Computing transitive closure the better way:

Warshall’s Algorithm procedure Warshall(M R : n by n zero-one matrix) W := M R for k:=1 to n for i:=1 to n for j:=1 to n w ij := w ij  (w ik  w kj ) { W now contains M R* }

Illustration of Warshall’s Algorithm

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  = {A 1, A 2, …, A m } of nonempty subsets of S which satisfies the following properties: –Every element of A is in one of the sets A i. –For all i, j  {1, 2, …, m}, if i  j then A i  A j =  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,,, … …,,-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

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  relation 2.The real numbers R under the  relation 3.The positive integers under the “divides” relation 4.Any set of sets under the  (subset) relation 5.The cartesian product Z  Z under the “( ,  )” relation R. (i.e. (x,y) R (z,w) if and only if x  z and y  w.)

Convention

Comparable and Incomparable Elements

Examples:

Total Orderings

Well-Ordered Sets

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

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 Elements

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

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

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