Slides for a Course Based on the Text Discrete Mathematics & Its Applications (5 th Edition) by Kenneth H. Rosen

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

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

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

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

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

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

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

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

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

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

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

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

§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

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

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

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

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

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

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

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

§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

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

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

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

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

§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

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

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

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

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

§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

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

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

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

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

§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

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

Slides for a Course Based on the Text Discrete Mathematics & Its Applications (5 th Edition) by Kenneth H. Rosen