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Lecture 14 Relations

Lecture 14 Relations. CSCI – 1900 Mathematics for Computer Science Spring 2014 Bill Pine. Lecture Introduction. Reading Kolman - Sections 4.1, 4.2, 4.4, 4.5 Cartesian Product Partitioning Relations Representing Relations as a Matrix Digraph Properties of Relations

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Lecture 14 Relations

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

  2. Lecture Introduction • Reading • Kolman - Sections 4.1, 4.2, 4.4, 4.5 • Cartesian Product • Partitioning • Relations • Representing Relations as a Matrix • Digraph • Properties of Relations • Equivalence Relations CSCI 1900

  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) CSCI 1900

  4. Cartesian Product of m Sets • If A1, A2, …, Am are nonempty sets, then the Cartesian Product of them is the set of all ordered m-tuples (a1, a2, …, am), where ai Ai, i = 1, 2, … m • Denoted A1 A2 …  Am = { (a1, a2, …, am) | ai Ai, i = 1, 2, … m } CSCI 1900

  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)} CSCI 1900

  6. Cardinality of Cartesian Product The Cardinality of the Cartesian Product equals the product of the cardinality of all sets involved sets: | A1 A2 …  Am | = | A1 |  | A2 |  …  | Am | CSCI 1900

  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} • A1 ={1, 3, 4} A2 ={2, 6} A3= {5} • P= {A1 , A2 , A3 } is a partition of A CSCI 1900

  8. Additional Partition Example • Let A={a, b, c, d, e, f, g, h} A1 ={a, b, c, d} A2 ={a, c, e, f, g, h} A3= {a, c, e, g} A4 ={b, d} A5 ={f, h} • Is P = {A1, A2 } a partition of A? • Is P = {A1, A5 } a partition of A? • Is P = {A3, A4 , A5} a partition of A? CSCI 1900

  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 CSCI 1900

  10. x R y 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 CSCI 1900

  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 BurfSnerfleR CSCI 2710 Ginger AyleR CSCI 2710 CSCI 1900

  12. Domain and Range • Example: A={1, 2, 3, 6} B={1, 3, 4} • Let Q be the relation, aRbif 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 CSCI 1900

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

  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 RC2 • C1 is a prerequisite for C2 • P = {(CSCI 2150, CSCI 3400), (CSCI 1710, CSCI 2910), (CSCI 2800, CSCI 2910), …} CSCI 1900

  15. Represent A Relation As A Matrix • Let: A = { a1, a2, a3, …, am} with |A| = mB = { b1, b2, b3, …, an} with |B| = n • With Q a relation from A to B • Represent Q as an m x n matrix, MQ , with mi j = 1 if (ai, bj)  Q = 0 if (ai, bj)  Q CSCI 1900

  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 CSCI 1900

  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 ai is related to aj, then draw an arrow from the vertex ai to the vertex aj • These arrows are called Arcs or Edges • The resulting pictorial is called a digraph(directed graph) CSCI 1900

  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 CSCI 1900

  19. 2 1 3 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) } CSCI 1900

  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 CSCI 1900

  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 CSCI 1900

  22. 1 2 4 3 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) } CSCI 1900

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

  24. 1 2 4 3 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) } CSCI 1900

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

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

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

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

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