Discrete Mathematics and its Applications 9/21/2018 University of Florida Dept. of Computer & Information Science & Engineering COT 3100 Applications of Discrete Structures Dr. Michael P. Frank Slides for a Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen A word about organization: Since different courses have different lengths of lecture periods, and different instructors go at different paces, rather than dividing the material up into fixed-length lectures, we will divide it up into “modules” which correspond to major topic areas and will generally take 1-3 lectures to cover. Within modules, we have smaller “topics”. Within topics are individual slides. 9/21/2018 (c)2001-2003, Michael P. Frank (c)2001-2002, Michael P. Frank

Rosen 5th ed., ch. 7 ~35 slides (in progress), ~2 lectures Module #21: Relations Rosen 5th ed., ch. 7 ~35 slides (in progress), ~2 lectures 9/21/2018 (c)2001-2003, Michael P. Frank

Binary Relations Let A, B be any sets. A binary relation R 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 R b 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} 9/21/2018 (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 by R :≡ {(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} = ≥ 9/21/2018 (c)2001-2003, Michael P. Frank

Inverse Relations Any binary relation R:A×B has an inverse relation R−1:B×A, defined by R−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  a eats b, then: b R−1 a  b is eaten by a. (Passive voice.) 9/21/2018 (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}. 9/21/2018 (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 “not reflexive”! 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! 9/21/2018 (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. 9/21/2018 (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… ? 9/21/2018 (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.” 9/21/2018 (c)2001-2003, Michael P. Frank

Functionality A relation R:A×B is functional if, for any aA, there is at most 1 b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 function R: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. 9/21/2018 (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 f without 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→B where 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. 9/21/2018 (c)2001-2003, Michael P. Frank

Composite Relations Let R:A×B, and S:B×C. Then the composite S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. 9/21/2018 (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 subset R  A1× … × An. The sets Ai are called the domains of R. The degree of R is n. R is functional in the domain Ai if it contains at most one n-tuple (…, ai ,…) for any value ai within domain Ai. 9/21/2018 (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×… 9/21/2018 (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 operator sC 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} 9/21/2018 (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). 9/21/2018 (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-tuples is defined by: 9/21/2018 (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} 9/21/2018 (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. 9/21/2018 (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). 9/21/2018 (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. 9/21/2018 (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 representation of the relation Likes:Boys×Girls relation is: 9/21/2018 (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 identical across diagonal Antisymmetric: all 1’s are across from 0’s 9/21/2018 (c)2001-2003, Michael P. Frank

Using Directed Graphs A directed graph or digraph G=(VG,EG) is a set VG of 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) Matrix representation MR: Graph rep. GR: Joe Susan Fred Mary Mark Sally Node set VG (black dots) 9/21/2018 (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 node has a self-loop Irreflexive: No node links to itself Symmetric: Every link is bidirectional Antisymmetric: No link is bidirectional These are asymmetric & non-antisymmetric These are non-reflexive & non-irreflexive 9/21/2018 (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 9/21/2018 (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 to b. (“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. 9/21/2018 (c)2001-2003, Michael P. Frank

Simple Transitive Closure Alg. A procedure to compute R* with 0-1 matrices. procedure transClosure(MR:rank-n 0-1 mat.) A := B := MR; for i := 2 to n begin A := A⊙MR; B := B  A {join} end return B {Alg. takes Θ(n4) time} {note A represents Ri, B represents } 9/21/2018 (c)2001-2003, Michael P. Frank

A Faster Transitive Closure Alg. procedure transClosure(MR:rank-n 0-1 mat.) A := MR; for i := 1 to log2 n begin A := A⊙(A+In); {A represents } end return A {Alg. takes only Θ(n3 log n) time!} 9/21/2018 (c)2001-2003, Michael P. Frank

Roy-Warshall Algorithm Uses only Θ(n3) operations! Procedure Warshall(MR : rank-n 0-1 matrix) W := MR for k := 1 to n for i := 1 to n for j := 1 to n wij := wij  (wik  wkj) return W {this represents R*} wij = 1 means there is a path from i to j going only through nodes ≤k 9/21/2018 (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. 9/21/2018 (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) 9/21/2018 (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]R is a function of a, any equivalence relation R can be defined using aRb :≡ “a and b have the same f value”, given f. 9/21/2018 (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, …} 9/21/2018 (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. 9/21/2018 (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, ≼). 9/21/2018 (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 minimal digraph: 2 4 8 3 6 1 5 9 7 10 9/21/2018 (c)2001-2003, Michael P. Frank

More to come… More slides on partial orderings to be added in the future… 9/21/2018 (c)2001-2003, Michael P. Frank