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Relations

Relations. Relations. Binary relation N-ary relations and relational database Representing relations Equivalence relations. Binary Relations. Let A , B be any two sets. A binary relation R from A to B , written (with signature) R : A ↔ B , is a subset of A × B .

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Relations

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  1. Relations Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  2. Relations • Binary relation • N-ary relations and relational database • Representing relations • Equivalence relations Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  3. Binary Relations • Let A, B be any two sets. • A binary relationR from A to B, written (with signature) R:A↔B, is a subset of A×B. • E.g., let <: N↔N :≡ {(n,m)| n < m} • The notation a Rb or 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)”, or “a relates to b (under relation R)”. • A binary relation R corresponds to a predicate function PR:A×B→{T,F} defined over the 2 sets A,B; e.g., “eats” :≡ {(a,b)| organism a eats food b} Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  4. 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. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen Example: < = {(a,b) | (a,b)<} = {(a,b) | ¬a<b} = ≥

  5. Relations on a Set • A (binary) relation from a set A to itself is called a relation on the set A. • E.g., the “<” relation from earlier was defined as a relation on the set N of natural numbers. • The identity relation IA on a set A is the set {(a,a)|aA}. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  6. Reflexivity, Symmetry & Antisymmetry • A relation R on A is reflexive if aA,aRa. • E.g., the relation ≥ :≡ {(a,b) | a≥b} is reflexive. • A binary relation R on A is symmetric, 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. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  7. Transitivity • A relation R is transitive iff (for all a,b,c)(a,b)R  (b,c)R→ (a,c)R. • A relation is intransitive if it is not transitive. • Examples: “is an ancestor of” is transitive. • “likes” is intransitive. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  8. 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) | b:aRb bSc} • Note function composition fg is an example. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  9. n-ary Relations • An n-ary relation R on sets A1,…,An, written R:A1,…,An, is a subsetR  A1× … × An. • The sets Ai are called the domains of R. • The degree of R is n. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  10. Relational Databases • A relational database is essentially an n-ary relation R. • A domain Ai is a primary key for the database if the relation R is functional in Ai. • A composite key for the database is a set of domains {Ai, Aj, …} such that R contains at most 1 n-tuple (…,ai,…,aj,…) for each composite value (ai, aj,…)Ai×Aj×… Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  11. Selection Operators • Let A be any n-ary domain A=A1×…×An, and let C:A→{T,F} be any condition (predicate) on elements (n-tuples) of A. • Then, the selection operatorsC is the operator that maps any (n-ary) relation R on A to the n-ary relation of all n-tuples from R that satisfy C. • i.e., RA,sC(R) = R{aA | sC(a) = T} Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  12. Selection Operator 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). Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  13. Projection Operators • Let A = A1×…×An be any n-ary domain, and let {ik}=(i1,…,im) be a sequence of indices all falling in the range 1 to n, • That is, where 1 ≤ ik ≤ n for all 1 ≤ k ≤ m. • Then the projection operator on n-tuplesis defined by: Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  14. Projection Example • 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} Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  15. 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. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  16. 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). Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  17. 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. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  18. 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: Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  19. é ù é ù é ù 1 1 0 ê ú ê ú ê ú 1 1 0 1 0 1 ê ú ê ú ê ú ê ê ê ú ú ú 1 0 ê ê ê ú ú ú 1 0 0 ë ë ë û û û Symmetric:all identicalacross diagonal Antisymmetric:all 1’s are acrossfrom 0’s Reflexive:all 1’s on diagonal Zero-One Reflexive, Symmetric • Terms: Reflexive,symmetric, and antisymmetric. • These relation characteristics are very easy to recognize by inspection of the zero-one matrix. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  20. 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). Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen Edge set EG(orange arrows) GR MR Joe Susan Fred Mary Mark Sally Node set VG(black dots)

  21.          Digraph Reflexive, Symmetric • It is extremely easy to recognize the reflexive/symmetric/antisymmetric properties by graph inspection. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen  Antisymmetric:No link isbidirectional Reflexive:Every nodehas a self-loop Symmetric:Every link isbidirectional

  22. 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. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  23. 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).” • “Integers a and b have the same residue modulo m.” (for a given m>1) Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  24. 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)is a function of a, any equivalence relation R canbe defined using aRb :≡ “a and b have the same f value”, given that f. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  25. 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, …} • “Integers a and b have the same residue modulo m.” (for a given m>1) • [a] = the set {…, a−2m, a−m, a, a+m, a+2m, …} Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  26. 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. Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

  27. Review: Relations • Binary relation • N-ary relations and relational database • Representing relations • Equivalence relations • Equivalent classes and partitions Based on Rosen, Discrete Mathematics & Its Applications. Prepared by (c)2001-2004, Michael P. Frank and Modified By Mingwu Chen

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