DISCRETE COMPUTATIONAL STRUCTURES CS 23022 Fall 2005.

DISCRETE COMPUTATIONAL STRUCTURES CS 23022 Fall 2005

CS 23022 OUTLINE 9.Matrices & Closures 10.Congruences 11.Counting Principles 12.Discrete Probability 13.Recurrence Relations 14.Algorithm Complexity 15.Graph Theory 16.Trees & Networks 17.Grammars & Languages 1.Sets 2.Logic 3.Proof Techniques 4.Algorithms 5.Integers & Induction 6.Relations & Posets 7.Functions 8.Boolean Algebra & Combinatorial Circuits

Learning Objectives  Learn about relations and their basic properties  Explore equivalence relations  Become aware of closures  Learn about posets  Explore how relations are used in the design of relational databases

Relations  Relations are a natural way to associate objects of various sets

Relations  R can be described in  Roster form  Set-builder form

Relations  Arrow Diagram  Write the elements of A in one column  Write the elements B in another column  Draw an arrow from an element, a, of A to an element, b, of B, if (a,b)  R  Here, A = {2,3,5} and B = {7,10,12,30} and R from A into B is defined as follows: For all a  A and b  B, a R b if and only if a divides b  The symbol → (called an arrow) represents the relation R

Relations

 Directed Graph  Let R be a relation on a finite set A  Let A = {a, b, c, d}  R= {(a.a), (a,b), (b,b), (b,c), (b,a), (b,d), (c,d)}  Describe R pictorially as follows:  For each element of A, draw a small or big dot and label the dot by the corresponding element of A  Draw an arrow from a dot labeled a, to another dot labeled, b, if a R b.  Resulting pictorial representation of R is called the directed graph representation of the relation R

Relations

 Directed graph (Digraph) representation of R  Each dot is called a vertex  If a vertex is labeled, a, then it is also called vertex a  An arc from a vertex labeled a, to another vertex, b is called a directed edge, or directed arc from a to b  The ordered pair (A, R) is a directed graph, or digraph, of the relation R, where each element of A is called a vertex of the digraph

Relations  Directed graph (Digraph) representation of R (Continued)  For vertices a and b, if a R b, a is adjacent to b and b is adjacent from a  Because (a, a)  R, an arc from a to a is drawn; because (a, b)  R, an arc is drawn from a to b. Similarly, arcs are drawn from b to b, b to c, b to a, b to d, and c to d  For an element a  A such that (a, a)  R, a directed edge is drawn from a to a. Such a directed edge is called a loop at vertex a

Relations  Directed graph (Digraph) representation of R (Continued)  Position of each vertex is not important  In the digraph of a relation R, there is a directed edge or arc from a vertex a to a vertex b if and only if a R b

Relations  Domain and Range of the Relation  Let R be a relation from a set A into a set B. Then R ⊆ A x B. The elements of the relation R tell which element of A is R-related to which element of B

Relations

 Let A = {1, 2, 3, 4} and B = {p, q, r}. Let R = {(1, q), (2, r ), (3, q), (4, p)}. Then R −1 = {(q, 1), (r, 2), (q, 3), (p, 4)}  To find R −1, just reverse the directions of the arrows  D(R) = {1, 2, 3, 4} = Im(R −1 ), Im(R) = {p, q, r} = D(R −1 )

Relations

 Constructing New Relations from Existing Relations

Relations If a R b and b S c, then a T c requires that D(S)  Q and Im(R)  Q

Relations  Example:  Consider the relations R and S as given in Figure 3.7.  The composition S ◦ R is given by Figure 3.8.

Relations

Example 3.1.26 continued

Relations

 Consider the relation R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 4), (4, 1), (2, 3), (3, 2)} on the set S = {1, 2, 3, 4, 5}.  The digraph is divided into three distinct blocks. From these blocks the subsets {1, 4}, {2, 3}, and {5} are formed. These subsets are pairwise disjoint, and their union is S.  A partition of a nonempty set S is a division of S into nonintersecting nonempty subsets.

Relations

 Example: Let A denote the set of the lowercase English alphabet. Let B be the set of lowercase consonants and C be the set of lowercase vowels. Then B and C are nonempty, B ∩ C = , and A = B ∪ C. Thus, {B, C} is a partition of A.  Let A be a set and let {A 1, A 2, A 3, A 4, A 5 } be a partition of A. Corresponding to this partition, a Venn diagram, can be drawn, Figure 3.13

Relations

Relations – Equivalence Classes Example Consider the equivalence relation discussed earlier, congruence modulo m To makes things specific, consider m=3. Then for any two integers, n and t, the relation n R t exists iff 3 divides n – t. The equivalence class [0] of 0 is: [0] =  …,-6,-3,0,3,6,9,12…  This is the set of all values of n where 3|n – 0, i.e n R 0, similarly The equivalence class [1] of 1is: [1] =  …,-8,-5,-2,1,4,7,10…  This is the set of all values of n where 3|n – 1, i.e n R 1, similarly The equivalence class [2] of 2is: [2] =  …,-7,-4,-1,2,5,8,11…  This is the set of all values of n where 3|n – 2, i.e n R 2 It can be shown that Z = [0]  [1]  [2] and P = {[0], [1], [2]} is a partition of Z Aside :[0] represents all elements n of Z where n/3 has a remainder = 0 [1] represents all elements n of Z where n/3 has a remainder = 1 [2] represents all elements n of Z where n/3 has a remainder = 2

Relations

 Let A = {a, b, c, d, e, f, g, h, i, j }. Let R be a relation on A such that the digraph of R is as shown in Figure 3.14.  Then a, b, c, d, e, f, c, g is a directed walk in R as a R b,b R c,c R d,d R e, e R f, f R c, c R g. Similarly, a, b, c, g is also a directed walk in R. In the walk a, b, c, d, e, f, c, g, the internal vertices are b, c, d, e, f, and c, which are not distinct as c repeats.  this walk is not a path. In the walk a, b, c, g, the internal vertices are b and c, which are distinct. Therefore, the walk a, b, c, g is a path.

Relations

Partially Ordered Sets

brown R green because brown green (r = m = n) garage R grate because garag grate (r = m < n) partial R partially because partial = partial and m > n

Partially Ordered Sets  Digraphs of Posets  Because any partial order is also a relation, a digraph representation of partial order may be given.  Example: On the set S = {a, b, c}, consider the relation R = {(a, a), (b, b), (c, c ), (a, b)}.  From the directed graph it follows that the given relation is reflexive and transitive.  This relation is also antisymmetric because there is a directed edge from a to b, but there is no directed edge from b to a. Again, in the graph there are two distinct vertices a and c such that there are no directed edges from a to c and from c to a.

Partially Ordered Sets  Digraphs of Posets  Let S = {1, 2, 3, 4, 6, 12}. Consider the divisibility relation on S, which is a partial order  A digraph of this poset is as shown in Figure 3.20

 Closed Path  On the set S = {a, b, c } consider the relation R = {(a, a), (b, b), (c, c ), (a, b), (b, c ), (c, a)}  The digraph of this relation is given in Figure 3.21  In this digraph, a, b, c, a form a closed path. Hence, the given relation is not a partial order relation

Consider the poset (S,≤) where S = {2,4,5,10,15,20} and the partial order ≤ is the divisibility relation. Note: a covers b in the divisibility relation iff a / b = p, where p is prime.

Partially Ordered Sets  Hasse Diagram  Let S = {1, 2, 3}. Then P(S) = { , {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, S}  Now (P(S),≤) is a poset, where ≤ denotes the set inclusion relation. The poset diagram of (P(S),≤) is shown in Figure 3.22

 Hasse Diagram  Let S = {1, 2, 3}. Then P(S) = { , {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, S}  (P(S),≤) is a poset, where ≤ denotes the set inclusion relation  Draw the digraph of this inclusion relation (see Figure 3.23). Place the vertex A above vertex B if B ⊂ A. Now follow steps (2), (3), and (4)

 Hasse Diagram  Consider the poset (S,≤), where S = {2, 4, 5, 10, 15, 20} and the partial order ≤ is the divisibility relation  In this poset, there is no element b ∈ S such that b  5 and b divides 5. (That is, 5 is not divisible by any other element of S except 5). Hence, 5 is a minimal element. Similarly, 2 is a minimal element

Partially Ordered Sets  Hasse Diagram  10 is not a minimal element because 2 ∈ S and 2 divides 10. That is, there exists an element b ∈ S such that b < 10. Similarly, 4, 15, and 20 are not minimal elements  2 and 5 are the only minimal elements of this poset. Notice that 2 does not divide 5. Therefore, it is not true that 2 ≤ b, for all b ∈ S, and so 2 is not a least element in (S,≤). Similarly, 5 is not a least element. This poset has no least element

Partially Ordered Sets  Hasse Diagram  There is no element b ∈ S such that b  15, b > 15, and 15 divides b. That is, there is no element b ∈ S such that 15 < b. Thus, 15 is a maximal element. Similarly, 20 is a maximal element.  10 is not a maximal element because 20 ∈ S and 10 divides 20. That is, there exists an element b ∈ S such that 10 < b. Similarly, 4 is not a maximal element. Figure 3.24

Partially Ordered Sets  Hasse Diagram  20 and 15 are the only maximal elements of this poset  10 does not divide 15, hence it is not true that b ≤ 15, for all b ∈ S, and so 15 is not a greatest element in (S,≤)  This poset has no greatest element Figure 3.24

Partially Ordered Sets – Topological Ordering Example: Consider a poset ({1,2,4,5,12,20},|) and it’s Hasse diagram. 2 1 4 5 2012 A topological ordering of the given poset is a linear ordering which preserves the partial ordering. This is an example : 1  5  2  4  20  12. Other valid linear orderings swap order of 5 and 2 or 12 and 20. 1  5  2  4  20  12 1  2  5  4  20  12 1  5  2  4  12  20 1  2  5  4  12  20 An algorithm for obtaining a topological ordering of P =(A,R) might be1)remove a minimal element from A; 2)output the element 3) repeat 1 and 2 until A = 

Non-distributive Lattice Note that a ∧ (b ∨ c ) = a ∧ 1 = a and 0 = 0 ∨ 0 = (a ∧ b) ∨ (a ∧ c ) Now, a ≠ 0 so this is not a distributive lattice

 Complement  Let D 30 denote the set of all positive divisors of 30. Then D 30 = {1, 2, 3, 5, 6, 10, 15, 30}.  (D 30,≤) is a poset where a ≤ b if and only if a divides b (≤ is the divisibility relation)  D 30,≤) is a poset where a ≤ b if and only if a divides b (≤ is the divisibility relation). Because 1 divides all elements of D 30, it follows that 1 ≤ m, for all m ∈ D 30. Therefore, 1 is the least element of this poset.

Partially Ordered Sets  Complement  Let a, b ∈ D 30. Let d = gcd{a, b} and m = lcm{a, b}. Now d | a and d | b. Hence, d ≤ a and d ≤ b.  This shows that d is a lower bound of {a, b}. Let c ∈ D 30 and c ≤ a, c ≤ b.  This shows that d is a lower bound of {a, b}. Let c ∈ D 30 and c ≤ a, c ≤ b. Then, c | a and c | b and because d = gcd{a, b}, it follows that c | d, so c ≤ d. Thus, d = glb{a, b}.

Partially Ordered Sets  Complement  Because all the positive divisors of a, b are also divisors of 30, d ∈ D 30, so d = a ∧ b  It can be shown that m ∈ D 30 and m = a ∨ b  Hence, (D 30,≤) is a lattice with the least element integer 1 and the greatest element 30

Partially Ordered Sets  Complement  For any element a ∈ D 30, (30/a) ∈ D 30  Using the properties of gcd and lcm, it can be shown that a ∧ (30/a) = 1 and a ∨ (30/a) = 30  10 ∧ (30/10) = gcd{10, 3} = 1 and 10 ∨ (30/10) = lcm{10, 3} = 30  Hence, 3 is a complement of 10 in this lattice

Application: Relational Database  A database is a shared and integrated computer structure that stores  End-user data; i.e., raw facts that are of interest to the end user;  Metadata, i.e., data about data through which data are integrated  A database can be thought of as a well-organized electronic file cabinet whose contents are managed by software known as a database management system; that is, a collection of programs to manage the data and control the accessibility of the data

Application: Relational Database  In a relational database system, tables are considered as relations  A table is an n-ary relation, where n is the number of columns in the tables  The headings of the columns of a table are called attributes, or fields, and each row is called a record  The domain of a field is the set of all (possible) elements in that column

Application: Relational Database  Each entry in the ID column uniquely identifies the row containing that ID  Such a field is called a primary key  Sometimes, a primary key may consist of more than one field

Application: Relational Database  Structured Query Language (SQL)  Information from a database is retrieved via a query, which is a request to the database for some information  A relational database management system provides a standard language, called structured query language (SQL)

Application: Relational Database  Structured Query Language (SQL)  An SQL contains commands to create tables, insert data into tables, update tables, delete tables, etc.  Once the tables are created, commands can be used to manipulate data into those tables.  The most commonly used command for this purpose is the select command. The select command allows the user to do the following:  Specify what information is to be retrieved and from which tables.  Specify conditions to retrieve the data in a specific form.  Specify how the retrieved data are to be displayed.

CSE 2353 OUTLINE 1.Sets 2.Logic 3.Proof Techniques 4.Integers and Induction 5.Relations and Posets 6.Functions 7.Counting Principles 8.Boolean Algebra

Learning Objectives  Learn about functions  Explore various properties of functions  Learn about sequences and strings  Become familiar with the representation of strings in computer memory  Learn about binary operations

Functions

 Every function is a relation  Therefore, functions on finite sets can be described by arrow diagrams. In the case of functions, the arrow diagram may be drawn slightly differently.  If f : A → B is a function from a finite set A into a finite set B, then in the arrow diagram, the elements of A are enclosed in ellipses rather than individual boxes.

Functions  To determine from its arrow diagram whether a relation f from a set A into a set B is a function, two things are checked:  Check to see if there is an arrow from each element of A to an element of B  This would ensure that the domain of f is the set A, i.e., D(f) = A  Check to see that there is only one arrow from each element of A to an element of B  This would ensure that f is well defined

Functions  Let A = {1,2,3,4} and B = {a, b, c, d} be sets  The arrow diagram in Figure 5.6 represents the relation f from A into B  Every element of A has some image in B  An element of A is related to only one element of B; i.e., for each a ∈ A there exists a unique element b ∈ B such that f (a) = b

Functions  Therefore, f is a function from A into B  The image of f is the set Im(f) = {a, b, d}  There is an arrow originating from each element of A to an element of B  D(f) = A  There is only one arrow from each element of A to an element of B  f is well defined

Functions  The arrow diagram in Figure 5.7 represents the relation g from A into B  Every element of A has some image in B  D(g ) = A  For each a ∈ A, there exists a unique element b ∈ B such that g(a) = b  g is a function from A into B

Functions  The image of g is Im(g) = {a, b, c, d} = B  There is only one arrow from each element of A to an element of B  g is well defined

Functions  Let h be the relation described by the arrow diagram in Figure 5.8  Every element of A has some image in B; i.e., there is an arrow originating from each element of A to an element of B. Therefore,  D(h) = A  However,element 1 has two images in B; i.e., there are two arrows originating from 1, one going to a and another going to b, so h is not well defined. Thus, the first condition of Definition 5.1.1 is satisfied,but the second one is not.  Therefore, h is not a function

Functions  The arrow diagram in Figure 5.9 represents a relation from A into B  Not every element of A has an image in B. For example, the element 4 has no image in B. In other words, there is no arrow originating from 4  Therefore, 4  D(k), so D(k)  A  This implies that k is not a function from A into B

Functions  Numeric Functions  If the domain and the range of a function are numbers, then the function is typically defined by means of an algebraic formula  Such functions are called numeric functions  Numeric functions can also be defined in such a way so that different expressions are used to find the image of an element

Functions

One-to-One, Onto and One-to-One Correspondence

Functions  Let A = {1,2,3,4} and B = {a, b, c, d}. Let f : A → B be a function such that the arrow diagram of f is as shown in Figure 5.10  The arrows from a distinct element of A go to a distinct element of B. That is, every element of B has at most one arrow coming to it.  If a 1, a 2 ∈ A and a 1 = a 2, then f(a 1 ) = f(a 2 ). Hence, f is one-one.  Each element of B has an arrow coming to it. That is, each element of B has a preimage.  Im(f) = B. Hence, f is onto B. It also follows that f is a one-to-one correspondence. Example 5.1.16

Functions  Let A = {1,2,3,4} and B = {a, b, c, d, e}  f : 1 → a, 2 → a, 3 → a, 4 → a  For this function the images of distinct elements of the domain are not distinct. For example 1  2, but f(1) = a = f(2).  Im(f) = {a}  B. Hence, f is neither one-one nor onto B. Example 5.1.18

Functions  Let A = {1,2,3,4} and B = {a, b, c, d, e}  f : 1 → a, 2 → b, 3 → d, 4 → e  For this function, the images of distinct elements of the domain are distinct. Thus, f is one-one. In this function, for the element c of B, the codomain, there is no element x in the domain such that f(x) = c ; i.e., c has no preimage. Hence, f is not onto B.

Functions

 Let A = {1,2,3,4}, B = {a, b, c, d, e},and C = {7,8,9}. Consider the functions f : A → B, g : B → C as defined by the arrow diagrams in Figure 5.14.  The arrow diagram in Figure 5.15 describes the function h = g ◦ f : A → C.

Functions In general, g  f ≠ f  g, even when both are defined. In other words, the composition of functions is not commutative. Example: f(n) = (-1) n, g(n) = 2n

Functions

Special Functions and Cardinality of a Set

If f is one-to-one, then f is left invertable. If f is onto, then f is right invertible. Example: f : , where f (x) = 9x + 5 is both one-to one and onto so it is left-invertible and right-invertible

Special Functions and Cardinality of a Set f | A’ is actually the same as f, except it’s domain is a subset of f ’s domain

Note: I n = {1,2,3,…,n}

Sequences and Strings

 m is the lower limit of the sum, n the upper limit of the sum, and a i the general term of the sum

 m is the lower limit of the product, n the upper limit of the product, and a i the general term of the product.

Sequences and Strings  Representing Strings into Computer Memory  A convenient way of storing a string into computer memory is to use an array.  Programming languages such as C++ and Java provide a data type to manipulate strings.  This data type includes algorithms to implement operations such as concatenation, finding the length of a string,determining whether a string is a substring in another string, and finding a substring into another string.

Binary Operations

 Let A be an alphabet. Let A + be the set of all nonempty strings on the alphabet A. If u, v ∈ A +,i.e., u and v are two nonempty strings on A,then the concatenation of u and v, uv ∈ A +. Thus, the concatenation of strings is a binary operation on A +. Also, by Theorem 5.3.23(ii), the concatenation operation is associative.  Hence, A + becomes a semigroup with respect to the concatenation operation.  Called a free semigroup generated by A.

Binary Operations  Let A * denote the set of all words including empty word λ. By Theorem 5.3.23(i), for all s ∈ A *, λs = s = sλ. Thus, A * becomes a monoid with identity 1 = λ.  This monoid is called a free monoid generated by A.  If A contains more than one element, then A * is a noncommutative monoid.

DISCRETE COMPUTATIONAL STRUCTURES CS 23022 Fall 2005.

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