1. Lecture 14 Relations CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine

2. CSCI 1900 Lecture 14 - 2 Lecture Introduction Reading –Rosen - Sections 9.1, 9.2, 9.3 Cartesian Product Partitioning Relations Representing Relations as a Matrix Digraph Properties of Relations Equivalence Relations

3. CSCI 1900 Lecture 14 - 3 Cartesian Product for 2 Sets If A and B are nonempty sets, the product set or Cartesian Product, A x B is the set of all ordered pairs (a,b) with a  A and b  B The ordered pair (a,b) is called a 2-tuple Example A={1,2,3} B={r,s} –A x B={ (1, r), (2, r), (3, r), (1, s), (2, s), (3, s) } Example Z x Z, set of all discrete points in the plane Note that (1,r)  (r,1)

4. CSCI 1900 Lecture 14 - 4 Cartesian Product of m Sets If A 1, A 2, …, A m are nonempty sets, then the Cartesian Product of them is the set of all ordered m-tuples (a 1, a 2, …, a m ), where a i  A i, i = 1, 2, … m Denoted A 1  A 2  …  A m = { (a 1, a 2, …, a m ) | a i  A i, i = 1, 2, … m }

5. CSCI 1900 Lecture 14 - 5 Cartesian Product Examples Let A = {1, 2, 3} and B = {a, b, c}, –A  B = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c), (3,a), (3,b), (3,c)} Let A = {1, 2, 3}, B = {a, b, c}, C = {T, F} –A  B  C= {(1,a,T), (1,b,T), (1,c,T), (2,a,T), (2,b,T), (2,c,T), (3,a,T), (3,b,T), (3,c,T), (1,a,F), (1,b,F), (1,c,F), (2,a,F), (2,b,F), (2,c,F), (3,a,F), (3,b,F), (3,c,F)}

6. CSCI 1900 Lecture 14 - 6 Cardinality of Cartesian Product The Cardinality of the Cartesian Product equals the product of the cardinality of all sets involved sets: | A 1  A 2  …  A m | = | A 1 |  | A 2 |  …  | A m |

7. CSCI 1900 Lecture 14 - 7 Partition A Partition P of a nonempty set A, is the set of nonempty subsets such that –Each element in A belongs to one set in P –Sets in P are disjoint Example: Let A={1, 2, 3, 4, 5, 6} –A 1 ={1, 3, 4} A 2 ={2, 6} A 3 = {5} –P = {A 1, A 2, A 3 } is a partition of A

8. CSCI 1900 Lecture 14 - 8 Additional Partition Example Let A={a, b, c, d, e, f, g, h} A 1 ={a, b, c, d} A 2 ={a, c, e, f, g, h} A 3 = {a, c, e, g} A 4 ={b, d} A 5 ={f, h} –Is P = {A 1, A 2 } a partition of A? –Is P = {A 1, A 5 } a partition of A? –Is P = {A 3, A 4, A 5 } a partition of A?

9. CSCI 1900 Lecture 14 - 9 Subsets of the Cartesian Product Many of the results of operations on sets produce subsets of the Cartesian Product set Relational database –Each column in a database table can be considered a set –Each row is an m-tuple of the elements from each column or set –No two rows should be alike Example –Employee ID, Last Name, Department, Salary –Database is a subset of ID x Name x Depart x Salary

10. CSCI 1900 Lecture 14 - 10 Relations A relation, R, is a subset of a Cartesian Product together with a property that defines whether a given m-tuple is a member of the relation or not Terminology: Relation R from A to B –R  A  B –Denoted by “x R y” where x  A and y  B x has a relation with y means: x does not have a relation with y x R y

11. CSCI 1900 Lecture 14 - 11 Relation Example Let –A be the set of all students, and –B be the set of all courses A relation R may be defined because certain courses are required by concentration Burf Snerfle R CSCI 2710 Ginger Ayle R CSCI 2710

12. CSCI 1900 Lecture 14 - 12 Domain and Range Example: A={1, 2, 3, 6} B={1, 3, 4} –Let Q be the relation, a R b if a < b –Q={ (1, 3), (1, 4), (2, 3), (2, 4), (3, 4) } Domain of Q, Dom(Q) ={1, 2, 3} –The set of elements in A that are related to some element in B Range of Q, Ran(Q) ={3, 4} –The set of elements in B that are related to some element of A

13. CSCI 1900 Lecture 14 - 13 Relations Across Same Set Relations may be defined upon a single set Terminology: Relation R on A R  A  A Consider the X-Y Plane you used to plot points

14. CSCI 1900 Lecture 14 - 14 Relation on a Single Set Example A is the set of all courses A relation P may be defined as the course is a prerequisite for Let P be the relation, C1 R C2 –C1 is a prerequisite for C2 P = {(CSCI 2150, CSCI 3400), (CSCI 1710, CSCI 2910), (CSCI 2800, CSCI 2910), …}

15. CSCI 1900 Lecture 14 - 15 Represent A Relation As A Matrix Let: A = { a 1, a 2, a 3, …, a m } with |A| = m B = { b 1, b 2, b 3, …, a n } with |B| = n With Q a relation from A to B Represent Q as an m x n matrix, M Q, with m i j = 1 if (a i, b j )  Q = 0 if (a i, b j )  Q

16. CSCI 1900 Lecture 14 - 16 Matrix Example Let: A = { 1, 2, 3 } B = { r, s } With the relation Q from A to B Q = { (1, r), (2, s), (3, r) } Then 1 0 M = 0 1 1 0

17. CSCI 1900 Lecture 14 - 17 Digraph of a Relation Let Q be a relation on A We can represent Q pictorially as follows –Each element of A is a circle called a Vertex –If a i is related to a j, then draw an arrow from the vertex a i to the vertex a j –These arrows are called Arcs or Edges The resulting pictorial is called a digraph (directed graph)

18. CSCI 1900 Lecture 14 - 18 Digraph of a Relation (cont) In-Degree of a vertex –Number of arcs pointing into a vertex Out-Degree of a vertex –Number of arcs pointing out from a vertex

19. CSCI 1900 Lecture 14 - 19 Representing a Relation Two representations of the same relation Q on A = {1, 2, 3} Q = { (1, 1), (1, 3), (2, 3), (3, 2), (3, 3) } 1 2 3 Node In Degree Out Degree 112 211 332

20. CSCI 1900 Lecture 14 - 20 Paths in a Digraph A walk from one node to the next, following the arrows is a Path –Vertex1, Edge1, Vertex2, Edge2, Vertex3, Edge3, Vertex4 –A vertex may occur multiple times in a path; an edge may occur only once Paths start and end at a vertex The number of edges in the path is its Length –The above example is length 3 A path that starts and ends at the same node is a cycle

21. CSCI 1900 Lecture 14 - 21 Properties of Relations Recall –If Q  A x A, then Q is a relation on A Relations on a set may have three properties –Reflexive –Symmetric –Transitive

22. CSCI 1900 Lecture 14 - 22 Reflexive A relation Q on set A, is reflexive if for all a  A, then (a, a)  Q Example: Let A={1, 2, 3, 4} Q={ (1,1), (1,4), (2,1), (2,2), (2,4), (3,3), (4,3), (4,4) } 1 2 3 4

23. CSCI 1900 Lecture 14 - 23 Reflexive( cont) A relation Q on set A, is irreflexive if no element relates to itself

24. CSCI 1900 Lecture 14 - 24 Symmetric A relation Q on set A, is symmetric –If (a, b)  Q then (b, a) is also  Q A = {1, 2, 3} Q = { (1, 1), (1, 2), (2, 1), (2, 3), (3, 2) } 1 2 3 4

25. CSCI 1900 Lecture 14 - 25 Symmetric ( cont) A relation Q on set A, is asymmetric if whenever (a, b)  Q, (b, a)  Q

26. CSCI 1900 Lecture 14 - 26 Transitive A relation Q on set A, is transitive if (a, b)  Q and (b, c)  Q then (a, c)  Q Example: Let A={ 1, 2, 3 } R={ (1, 2), (2, 3), (1, 3) } 1 2 3 4

27. CSCI 1900 Lecture 14 - 27 Equivalence Relations A relation Q on a set A is an equivalence relation if the relation is –Reflexive –Symmetric –Transitive

28. CSCI 1900 Lecture 14 - 28 Key Concepts Summary Cartesian Product Partitioning Relations Representing Relations as a Matrix Digraph Properties of Relations Equivalence Relations