1 / 39

Discrete Mathematics: Binary Relations, Complementary Relations, Inverse Relations, Relations on a Set, Reflexivity, Sym

This module covers various topics related to binary relations in discrete mathematics, including complementary relations, inverse relations, reflexivity, symmetry and antisymmetry, transitivity, totality, functionality, and lambda notation. It also explores relations on a set and their properties.

mathildaj
Download Presentation

Discrete Mathematics: Binary Relations, Complementary Relations, Inverse Relations, Relations on a Set, Reflexivity, Sym

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. University of FloridaDept. of Computer & Information Science & EngineeringCOT 3100Applications of Discrete StructuresDr. Michael P. Frank Slides for a Course Based on the TextDiscrete Mathematics & Its Applications (5th Edition)by Kenneth H. Rosen (c)2001-2003, Michael P. Frank

  2. Module #21:Relations Rosen 5th ed., ch. 7 ~35 slides (in progress), ~2 lectures (c)2001-2003, Michael P. Frank

  3. Binary Relations • Let A, B be any sets. A binary relationR from A to B, written (with signature) R:A×B, or R:A,B, is (can be identified with) a subset of the set A×B. • E.g., let <: N↔N :≡ {(n,m)| n < m} • The notation a Rb or aRb means that (a,b)R. • E.g., a <b means (a,b) < • If aRb we may say “a is related to b (by relation R)”, • or just “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., predicate “eats” :≡ {(a,b)| organism a eats food b} (c)2001-2003, Michael P. Frank

  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. Example:< = {(a,b) | (a,b)<} = {(a,b) | ¬a<b} = ≥ (c)2001-2003, Michael P. Frank

  5. 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} = >. • E.g., if R:People→Foods is defined by a R b  aeatsb, then: bR−1a  bis eaten bya. (Passive voice.) (c)2001-2003, Michael P. Frank

  6. 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 (binary) identity relation IA on a set A is the set {(a,a)|aA}. (c)2001-2003, Michael P. Frank

  7. Reflexivity • A relation R on A is reflexive if aA,aRa. • E.g., the relation ≥ :≡ {(a,b) | a≥b} is reflexive. • A relation R is irreflexive iff its complementary relation R is reflexive. • Example: < is irreflexive, because ≥ is reflexive. • Note “irreflexive” does NOT mean “notreflexive”! • For example: the relation “likes” between people is not reflexive, but it is not irreflexive either. • Since not everyone likes themselves, but not everyone dislikes themselves either! (c)2001-2003, Michael P. Frank

  8. 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, (a,b)R→ (b,a)R. • Examples: < is antisymmetric, “likes” is not. • Exercise: prove this definition of antisymmetric is equivalent to the statement that RR−1  IA. (c)2001-2003, Michael P. Frank

  9. 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 iff it is not transitive. • Some examples: • “is an ancestor of” is transitive. • “likes” between people is intransitive. • “is located within 1 mile of” is… ? (c)2001-2003, Michael P. Frank

  10. Totality • A relation R:A×B is total if for every aA, there is at least one bB such that (a,b)R. • If R is not total, then it is called strictly partial. • A partial relation is a relation that might be strictly partial. (Or, it might be total.) • In other words, all relations are considered “partial.” (c)2001-2003, Michael P. Frank

  11. Functionality • A relation R:A×B is functionalif, for any aA, there is at most 1bB such that (a,b)R. • “R is functional”  aA: ¬b1≠b2B: aRb1  aRb2. • Iff R is functional, then it corresponds to a partial function R:A→B • where R(a)=b  aRb; e.g. • E.g., The relation aRb :≡ “a + b = 0” yields the function −(a) = b. • R is antifunctional if its inverse relation R−1 is functional. • Note: A functional relation (partial function) that is also antifunctional is an invertible partial function. • R is a total functionR:A→B if it is both functional and total, that is, for any aA, there is exactly 1 b such that (a,b)R. I.e., aA: ¬!b: aRb. • If R is functional but not total, then it is a strictly partial function. • Exercise: Show that iff R is functional and antifunctional, and both it and its inverse are total, then it is a bijective function. (c)2001-2003, Michael P. Frank

  12. Lambda Notation • The lambda calculus is a formal mathematical language for defining and operating on functions. • It is the mathematical foundation of a number of functional (recursive function-based) programming languages, such as LISP and ML. • It is based on lambda notation, in which “λa: f(a)” is a way to denote the function fwithout ever assigning it a specific symbol. • E.g., (λx. 3x2+2x) is a name for the function f:R→R where f(x)=3x2+2x. • Lambda notation and the “such that” operator “” can also be used to compose an expression for the function that corresponds to any given functional relation. • Let R:A×B be any functional relation on A,B. • Then the expression (λa: b aRb) denotes the function f:A→Bwhere f(a) = b iff aRb. • E.g., If I write:f :≡(λa: b  a+b = 0),this is one way of defining the function f(a)=−a. (c)2001-2003, Michael P. Frank

  13. 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 that function composition fg is an example. • Exer.: Prove that R:A×A is transitive iff RR = R. • The nth power Rn of a relation R on a set A can be defined recursively by:R0 :≡ IA; Rn+1 :≡ RnR for all n≥0. • Negative powers of R can also be defined if desired, by R−n:≡ (R−1)n. (c)2001-2003, Michael P. Frank

  14. §7.2: n-ary Relations • An n-ary relation R on sets A1,…,An,written (with signature) R:A1×…×An or R:A1,…,An, is simply a subsetR  A1× … × An. • The sets Ai are called the domains of R. • The degree of R is n. • R is functional in the domain Aiif it contains at most one n-tuple (…, ai ,…) for any value ai within domain Ai. (c)2001-2003, Michael P. Frank

  15. Relational Databases • A relational database is essentially just 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×… (c)2001-2003, Michael P. Frank

  16. 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) = {aR | sC(a) = T} (c)2001-2003, Michael P. Frank

  17. 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). (c)2001-2003, Michael P. Frank

  18. 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: (c)2001-2003, Michael P. Frank

  19. 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 (a database of cars) to obtain a list of the model/color combinations available. {ik} (c)2001-2003, Michael P. Frank

  20. 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, and C here can also be sequences of elements (across multiple fields), not just single elements. (c)2001-2003, Michael P. Frank

  21. 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). (c)2001-2003, Michael P. Frank

  22. §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. (c)2001-2003, Michael P. Frank

  23. Using Zero-One Matrices • To represent a binary relation R:A×B by an |A|×|B| 0-1 matrix MR = [mij], let mij = 1 iff (ai,bj)R. • E.g., Suppose Joe likes Susan and Mary, Fred likes Mary, and Mark likes Sally. • Then the 0-1 matrix representationof the relationLikes:Boys×Girlsrelation is: (c)2001-2003, Michael P. Frank

  24. Zero-One Reflexive, Symmetric • Terms: Reflexive, non-reflexive, irreflexive,symmetric, asymmetric, 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 Irreflexive:all 0’s on diagonal Symmetric:all identicalacross diagonal Antisymmetric:all 1’s are acrossfrom 0’s (c)2001-2003, Michael P. Frank

  25. 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) Graph rep. GR: Matrix representation MR: Joe Susan Fred Mary Mark Sally Node set VG(black dots) (c)2001-2003, Michael P. Frank

  26. 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 These are asymmetric & non-antisymmetric These are non-reflexive & non-irreflexive (c)2001-2003, Michael P. Frank

  27. §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 (c)2001-2003, Michael P. Frank

  28. 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 itself 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. (c)2001-2003, Michael P. Frank

  29. 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,B represents } (c)2001-2003, Michael P. Frank

  30. A Faster Transitive Closure Alg. proceduretransClosure(MR:rank-n 0-1 mat.) A := MR; fori := 1 to log2n beginA := A⊙(A+In); {A represents }endreturn A {Alg. takes only Θ(n3 log n) time!} (c)2001-2003, Michael P. Frank

  31. 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 (c)2001-2003, Michael P. Frank

  32. §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. (c)2001-2003, Michael P. Frank

  33. 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) (c)2001-2003, Michael P. Frank

  34. 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 can be defined using aRb :≡ “a and b have the same f value”, given f. (c)2001-2003, Michael P. Frank

  35. 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, …} (c)2001-2003, Michael P. Frank

  36. Partitions • A partition of a set A is the set of all the equivalence classes {A1, A2, … } for some equivalence relation on A. • The Ai’s are all disjoint and their union = A. • They “partition” the set into pieces. Within each piece, all members of that set are equivalent to each other. (c)2001-2003, Michael P. Frank

  37. §7.6: Partial Orderings • A relation R on A is called a partial ordering or partial order iff it is reflexive, antisymmetric, and transitive. • We often use a symbol looking something like ≼ (or analogous shapes) for such relations. • Examples: ≤, ≥ on real numbers, ,  on sets. • Another example: the divides relation | on Z+. • Note it is not necessarily the case that either a≼b or b≼a. • A set A together with a partial order ≼ on A is called a partially ordered set or poset and is denoted (A, ≼). (c)2001-2003, Michael P. Frank

  38. Posets as Noncyclical Digraphs • There is a one-to-one correspondence between posets and the reflexive+transitive closures of noncyclical digraphs. • Noncyclical: Containing no directed cycles. • Example: consider the poset ({0,…,10},|) • Its minimaldigraph: 2 4 8 6 3 1 0 9 5 10 7 (c)2001-2003, Michael P. Frank

  39. More to come… • More slides on partial orderings to be added in the future… (c)2001-2003, Michael P. Frank

More Related