1. Chapter 4 Relations and Digraphs
CSCI 115
Chapter 4
Relations and Digraphs

2. §4.1 Product Sets and Partitions
CSCI 115
§4.1
Product Sets and Partitions

3. §4.1 – Product Sets and Partitions
Ordered pair
Cartesian Product
Theorem 4.1.1
For any 2 finite non-empty sets A and B, |A x B| = |A||B|

4. §4.1 – Product Sets and Partitions
A partition (quotient set) of A is a set P of nonempty subsets of A such that:
Each element of A belongs to a set in P
If A1 and A2 are elements of P, then A1A2 = {}
Each element of P is called a cell or a block

5. §4.2 Relations and Digraphs
CSCI 115
§4.2
Relations and Digraphs

6. §4.2 – Relations and Digraphs
Relation from A to B
Subset of A x B
Relation on A
Subset of A x A

7. §4.2 – Relations and Digraphs
Sets arising from relations
Domains
Domain of relation
Ranges
Range of relation
Range of an element
Range of a subset

8. §4.2 – Relations and Digraphs
Theorem 4.2.1
Let R be a relation from A to B, and let A1 and A2 be subsets of A. Then:
If A1  A2, then R(A1)  R(A2)
R(A1  A2) = R(A1)  R(A2)
R(A1  A2)  R(A1)  R(A2)
Theorem 4.2.2
Let R and S be Relations from A to B. If R(a) = S(a)  a  A, then R = S.

9. §4.2 – Relations and Digraphs
The matrix of a relation
If A = {a1, a2, …, am} and B = {b1, b2, …, bn}, then a relation R from A to B can be represented by an m x n boolean matrix (MR) constructed as follows: mij = {
1 if (ai, bj)  R 0 if (ai, bj)  R

10. §4.2 – Relations and Digraphs
The Digraph of a relation
Let R be a relation on A. The Digraph of R is constructed as follows:
Draw a circle for each element in A, and label the circles accordingly (these are called vertices)
Draw an arrow from ai to aj iff aiRaj (these are called edges)
In-degrees and out-degrees

11. §4.2 – Relations and Digraphs
Relation Restriction
Let R be a relation on a set A, with B  A. The restriction of R to B is R  (B x B).

12. §4.3 Paths in Relations and Digraphs
CSCI 115
§4.3
Paths in Relations and Digraphs

13. §4.3 – Paths in Relations and Digraphs
Path of length n from a to b:
aRx1, x1Rx2, x2Rx3, …, xn-1Rb
Geometric path in digraph
Cycle: A path that begins and ends at the same vertex

14. §4.3 – Paths in Relations and Digraphs
New relations from paths
Rn: xRny iff there is a path of length n from x to y
R: xRy iff there is any path from x to y
Connectivity Relation
R*: xR*y iff there is a path from x to y, or x = y
Reachability Relation

15. §4.3 – Paths in Relations and Digraphs
Theorem 4.3.1
If R is a relation on A = {a1, a2, …, an}, then MR2 = MR  MR = ((MR))2
Theorem 4.3.2
For n  2 and R a relation on a finite set A, we have MRn = MR  MR...  MR (n factors)

16. §4.3 – Paths in Relations and Digraphs
Path composition
 is used as a variable for paths
Assume we have:
1: a, x1, x2, …, xn-1, b
2: b, y1, y2, …, ym-1, c
Then, we can compose 1 and 2 as follows:
21: a, x1, x2, …, xn-1, b, y1, y2, …, ym-1, c
21 is a path of length n + m
12 does not make sense

17. §4.4 Properties of Relations
CSCI 115
§4.4
Properties of Relations

18. §4.4 – Properties of Relations
A relation R on a set A is:
Reflexive iff (a,a)  R a  A
Irreflexive iff (a,a)  R a  A
Symmetric iff (a,b)  R  (b,a)  R
Asymmetric iff (a,b)  R  (b,a)  R
Antisymmetric iff (a,b)  R and (b,a)  R  a = b
Transitive iff (a,b)  R and (b,c)  R  (a,c)  R

19. §4.4 – Properties of Relations
Theorem 4.4.1
A relation R is transitive iff it satisfies the following property: If there is a path of length greater than 1 from vertex a to vertex b, there is a path of length 1 from a to b (i.e. aRb). In other words, R is transitive iff Rn  R  n > 1.

20. §4.4 – Properties of Relations
Theorem 4.4.2
Let R be a relation on a set A. Then:
Reflexivity of R means a  R(a)  a  A
Symmetry of R means a  R(b) iff b  R(a)
Transitivity of R means b  R(a) and c  R(b) implies c  R(a)

21. §4.4 – Properties of Relations
Other properties:
The digraph of a reflexive relation has a cycle of length 1 at every vertex
The digraph of an irreflexive relation has no cycles of length 1
If R is reflexive then the Domain(R) = Range(R) = A
The matrix of a reflexive relation has all 1s on the diagonal
The matrix of an irreflexive relation has all 0s on the diagonal
The matrix of a symmetric relation has mij = mji i,j (MR = MRT)
The matrix of an asymmetric relation meets the following 2 criteria:
mij = 1  mji = 0 AND the diagonal must be all 0s
The matrix of an antisymmetric relation meets the following criteria:
If ij, then mij = 0 or mji = 0
The Matrix of a transitive relation meets the following criteria:
MR must have a 1 everywhere ((MR))2 has a 1

22. §4.4 – Properties of Relations
Graph of a symmetric relation
Undirected edge
Adjacent vertices
Connected graphs
Disconnected graphs

23. §4.5 Equivalence Relations
CSCI 115
§4.5
Equivalence Relations

24. §4.5 – Equivalence Relations
A relation R on a set A is an equivalence relation if it is reflexive, symmetric, and transitive

25. §4.5 – Equivalence Relations
Theorem 4.5.1
Let P be a partition of a set A such that P = {A1, A2, …, An}. Define a relation R on A as follows: aRb iff aAi and bAi (i  {1, 2, …, n}). Then R is an equivalence relation on A.
Here R is called the equivalence relation determined by P.

26. §4.5 – Equivalence Relations
Lemma 4.5.1
Let R be an equivalence relation on A. Let a, b  A. Then aRb iff R(a) = R(b).
Theorem 4.5.2
Let R be an equivalence relation on A, and let P be the collection of all distinct relative sets R(a) for a in A. Then P is a partition of A, and R is the equivalence relation determined by P.
Note: When R is an equivalence relation on A, the sets R(a) are called equivalence classes. The partition constructed in Theorem is denoted A/R.

27. §4.5 – Equivalence Relations
Procedure to determine A/R (A finite or countable):
Choose any aA and find R(a).
If R(a)  A, choose bA, bR(a) and find R(b).
If A is not the union of the computed equivalence classes, choose xA such that x is not in any of the equivalence classes, and find R(x).
Repeat step 3 until A is accounted for, or a pattern emerges which describes the equivalence classes.
A/R is the partition formed by using each equivalence class as a cell.

28. §4.6 Data Structures for Relations and Digraphs
CSCI 115
§4.6
Data Structures for Relations and Digraphs

29. §4.6 – Data Structures for Relations and Digraphs
Linked list data structure
Storage cell
Data
Pointer
Implementations
Structures
Arrays
We will use the array implementation

30. §4.6 – Data Structures for Relations and Digraphs
Implementation 1 – Start, Tail, Head, Next
Fill in tail and head arrays
Fill in start variable
Fill in next array
The tail, head and next arrays have as many elements as the digraph has edges
Start is a simple variable (not an array)

31. §4.6 – Data Structures for Relations and Digraphs
Implementation 2 – Vert, Tail, Head, Next
Fill in tail and head arrays
Fill in next array (within vertex)
Fill in vert array. The data in vert is the first subscript in tail that corresponds to the current vertex
The tail, head and next arrays have as many elements as the digraph has edges
The vert array has as many elements as the digraph has vertices