Relations

1. Relations Rosen 5th ed., ch. 7

2. Relations • Relationships between elements of sets occur very often. • (Employee, Salary) • (Students, Courses, GPA) • We use ordered pairs (or n-tuples) of elements from the sets to represent relationships.

3. Binary Relations • A binary relationR from set A to set B, written R:A↔B, is a subset of A×B. • E.g., let <: N↔N :≡ {(n,m)| n < m} • The notation aRb means (a,b)R. • E.g., a <b means (a,b) < • If aRb we may say “a is related to b (by relation R)”

4. Example A: {students at UNR}, B: {courses offered at UNR} R: “relation of students enrolled in courses” (Jason, CS365), (Mary, CS201) are in R If Mary does not take CS365, then (Mary, CS365) is not in R! If CS480 is not being offered, then (Jason, CS480), (Mary, CS480) are not in R!

5. Complementary Relations • Let R:A↔B be any binary relation. • Then, R:A↔B, the complement of R, is the binary relation defined byR :≡ {(a,b) | (a,b)R} = (A×B) − R Note this is just R if the universe of discourse is U = A×B; thus the name complement. • Note the complement of R is R. Example: < = {(a,b) | (a,b)<} = {(a,b) | ¬a<b} = ≥

6. Inverse Relations • Any binary relation R:A↔B has an inverse relation R−1:B↔A, defined byR−1 :≡ {(b,a) | (a,b)R}. E.g., <−1 = {(b,a) | a<b} = {(b,a) | b>a} = >.

7. Functions as Relations • A function f:AB is a relation from A to B • A relation from A to B is not always a function f:AB (e.g., relations could be one-to-many) • Relations are generalizations of functions!

8. Relations on a Set • A (binary) relation from a set A to itself is called a relation on the set A. A: {1,2,3,4} R={(a,b) | a divides b}

9. Example How many relations are there on a set A with n elements?

10. Reflexivity • A relation R on A is reflexive if aA,aRa. • E.g., the relation ≥ :≡ {(a,b) | a≥b} is reflexive. • Is the “divide” relation on the set of positive integers reflexive?

11. Symmetry & Antisymmetry • A binary relation R on A is symmetric iff R = R−1, that is, if (a,b)R ↔ (b,a)R. • E.g., = (equality) is symmetric. < is not. • “is married to” is symmetric, “likes” is not. • A binary relation R is antisymmetric if (a,b)R → (b,a)R. • < is antisymmetric, “likes” is not.

12. Transitivity • A relation R is transitive iff (for all a,b,c) (a,b)R  (b,c)R→ (a,c)R. • Examples: • “is an ancestor of” is transitive. • Is the “divides” relation on the set of positive integers transitive?

13. Combining Relations • Two relations can be combined in a similar way to combining two sets.

14. Composite Relations • Let R:A↔B, and S:B↔C. Then the compositeSR of R and S is defined as: SR = {(a,c) | aRb bSc} • Function composition fg is an example. • The nth power Rn of a relation R on a set A can be defined recursively by: R1 :≡ R; Rn+1 :≡ RnR for all n≥0.

15. Example R is a relation from {1,2,3} to {1,2,3,4} R ={(1,1),(1,4),(2,3),(3,1),(3,4)} S is a relation from {1,2,3,4} to {0,1,2} S = {(1,0),(2,0),(3,1),(3,2),(4,1)} RS = {(1,0),(1,1),(2,1),(2,2),(3,0),(3,1)}

16. §7.2: n-ary Relations • An n-ary relation R on sets A1,…,An, written R:A1,…,An, is a subsetR  A1× … × An. • The degree of R is n. • Example: R consists of 5-tuples (A,N,S,D,T) A: airplane flights, N: flight number, S: starting point, D: destination, T: departure time

17. Databases • The time required to manipulate information in a database depends in how this information is stored. • Operations: add/delete, update, search, combine etc. • Various methods for representing databases have been developed. • We will discuss the “relational model”.

18. Relational Databases • A database consists of records, which are n-tuples, made up of fields. • A relational database represents records as an n-ary relation R. (STUDENT_NAME, ID, MAJOR, GPA) • Relations are also called “tables” (e.g., displayed as tables often)

19. Relational Databases • A domain Ai of an n-ary relation is called primary key when no two n-tuples have the same value on this domain (e.g., ID) • A composite key is a subset of domains {Ai, Aj, …} such that an n-tuple (…,ai,…,aj,…) is determined uniquely for each composite value (ai, aj,…)Ai×Aj×…

20. Selection Operator • Let R be an n-ary relation and C a condition that elements in R may satisfy. • The selection operatorsC maps any (n-ary) relation R on A to the n-ary relation of all n-tuples from R that satisfy C.

21. Example • Suppose we have a domain A = StudentName × Standing × SocSecNos • Suppose we define a certain condition on A, UpperLevel(name,standing,ssn) :≡ [(standing = junior) (standing = senior)] • Then, sUpperLevel is the selection operator that takes any relation R on A (database of students) and produces a relation consisting of just the upper-level classes (juniors and seniors).

22. Projection Operators • The projection operator maps the n-tuple to the m-tuple • In other words, it deletes n-m of the components of an n-tuple.

23. Example Note that fewer rows may result when a projection is applied !

24. More Examples • Suppose we have a ternary (3-ary) domain Cars=Model×Year×Color. (note n=3). • Consider the index sequence {ik}= 1,3. (m=2) • Then the projection P simply maps each tuple (a1,a2,a3) = (model,year,color) to its image: • This operator can be usefully applied to a whole relation RCars (database of cars) to obtain a list of model/color combinations available. {ik}

25. Join Operator • Puts two relations together to form a sort of combined relation. • If the tuple (A,B) appears in R1, and the tuple (B,C) appears in R2, then the tuple (A,B,C) appears in the join J(R1,R2). • A, B, C can also be sequences of elements rather than single elements.

26. Example

27. Join Example • Suppose R1 is a teaching assignment table, relating Professors to Courses. • Suppose R2 is a room assignment table relating Courses to Rooms,Times. • Then J(R1,R2) is like your class schedule, listing (professor,course,room,time).

28. SQL Example SELECT Departure_Time FROM Flights WHERE destination=“Detroit” Find projection P5 of the selection of 5-tuples that satisfy the constraint “destination=Detroit”

29. §7.3: Representing Relations • Some ways to represent n-ary relations: • With an explicit list or table of its tuples. • With a function from the domain to {T,F}. • Or with an algorithm for computing this function. • Some special ways to represent binary relations: • With a zero-one matrix. • With a directed graph.

30. Using Zero-One Matrices • To represent a relation R by a matrix MR = [mij], let mij = 1 if (ai,bj)R, else 0. • E.g., Joe likes Susan and Mary, Fred likes Mary, and Mark likes Sally. • The 0-1 matrix representationof that “Likes”relation:

31. Zero-One Reflexive, Symmetric • Terms: Reflexive, non-Reflexive, symmetric, and antisymmetric. • These relation characteristics are very easy to recognize by inspection of the zero-one matrix. any-thing any-thing anything anything any-thing any-thing Reflexive:all 1’s on diagonal Non-reflexive:some 0’s on diagonal Symmetric:all identicalacross diagonal Antisymmetric:all 1’s are acrossfrom 0’s

32. Using Directed Graphs • A directed graph or digraphG=(VG,EG) is a set VGof vertices (nodes) with a set EGVG×VG of edges (arcs,links). Visually represented using dots for nodes, and arrows for edges. Notice that a relation R:A↔B can be represented as a graph GR=(VG=AB, EG=R). Edge set EG(blue arrows) GR MR Joe Susan Fred Mary Mark Sally Node set VG(black dots)

33. Digraph Reflexive, Symmetric It is extremely easy to recognize the reflexive/irreflexive/ symmetric/antisymmetric properties by graph inspection.            Reflexive:Every nodehas a self-loop Irreflexive:No nodelinks to itself Symmetric:Every link isbidirectional Antisymmetric:No link isbidirectional Asymmetric, non-antisymmetric Non-reflexive, non-irreflexive

34. §7.4: Closures of Relations • For any property X, the “X closure” of a set A is defined as the “smallest” superset of A that has the given property. • The reflexive closure of a relation R on A is obtained by adding (a,a) to R for each aA. I.e.,it is R  IA • The symmetric closure of R is obtained by adding (b,a) to R for each (a,b) in R. I.e., it is R  R−1 • The transitive closure or connectivity relation of R is obtained by repeatedly adding (a,c) to R for each (a,b),(b,c) in R. • I.e., it is

35. Paths in Digraphs/Binary Relations • A path of length n from node a to b in the directed graph G (or the binary relation R) is a sequence (a,x1), (x1,x2), …, (xn−1,b) of n ordered pairs in EG (or R). • An empty sequence of edges is considered a path of length 0 from a to a. • If any path from a to b exists, then we say that a is connected tob. (“You can get there from here.”) • A path of length n≥1 from a to a is called a circuit or a cycle. • Note that there exists a path of length n from a to b in R if and only if (a,b)Rn.

36. Simple Transitive Closure Alg. A procedure to compute R* with 0-1 matrices. proceduretransClosure(MR:rank-n 0-1 mat.) A := B := MR; fori := 2 to nbeginA := A⊙MR; B := B A {join}endreturn B {Alg. takes Θ(n4) time} {note A represents Ri}

37. Roy-Warshall Algorithm • Uses only Θ(n3) operations! Procedure Warshall(MR : rank-n 0-1 matrix) W := MR fork := 1 tonfori := 1 tonforj := 1 tonwij := wij (wik  wkj)return W {this represents R*} wij = 1 means there is a path from i to j going only through nodes ≤k

38. §7.5: Equivalence Relations • An equivalence relation (e.r.) on a set A is simply any binary relation on A that is reflexive, symmetric, and transitive. • E.g., = itself is an equivalence relation. • For any function f:A→B, the relation “have the same f value”, or =f :≡ {(a1,a2) | f(a1)=f(a2)} is an equivalence relation, e.g., let m=“mother of” then =m = “have the same mother” is an e.r.

39. Equivalence Relation Examples • “Strings a and b are the same length.” • “Integers a and b have the same absolute value.” • “Real numbers a and b have the same fractional part (i.e., a− b Z).”

40. Equivalence Classes • Let R be any equiv. rel. on a set A. • The equivalence class of a, [a]R :≡ { b | aRb } (optional subscript R) • It is the set of all elements of A that are “equivalent” to a according to the eq.rel. R. • Each such b (including a itself) is called a representative of [a]R. • Since f(a)=[a]Ris a function of a, any equivalence relation R be defined using aRb :≡ “a and b have the same f value”, given that f.

41. Equivalence Class Examples • “Strings a and b are the same length.” • [a] = the set of all strings of the same length as a. • “Integers a and b have the same absolute value.” • [a] = the set {a, −a} • “Real numbers a and b have the same fractional part (i.e., a − b Z).” • [a] = the set {…, a−2, a−1, a, a+1, a+2, …}

42. Partitions • A partition of a set A is the set of all the equivalence classes {A1, A2, … } for some e.r. on A. • The Ai’s are all disjoint and their union = A. • They “partition” the set into pieces. Within each piece, all members of the set are equivalent to each other.