DISCRETE COMPUTATIONAL STRUCTURES

DISCRETE COMPUTATIONAL STRUCTURES
CSE 2353 Fall 2005

CSE 2353 OUTLINE Relations and Posets Sets Logic Proof Techniques
Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra

Learn about relations and their basic properties
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 Discrete Mathematical Structures: Theory and Applications

Relations are a natural way to associate objects of various sets
Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Relations Discrete Mathematical Structures: Theory and Applications

Relations Directed Graph Let R be a relation on a finite set A
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 Discrete Mathematical Structures: Theory and Applications

Directed graph (Digraph) representation of 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) a directed graph, or digraph, of the relation R, where each element of A is a called a vertex of the digraph Discrete Mathematical Structures: Theory and Applications

Directed graph (Digraph) representation of R (Continued)
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 Discrete Mathematical Structures: Theory and Applications

Directed graph (Digraph) representation of R (Continued)
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 Let A ={a ,b ,c ,d} and let R be the relation defined by the following set: R = {(a ,a ), (a ,b ), (b ,b ), (b ,c ), (b ,a ), (b ,d ), (c ,d )} Discrete Mathematical Structures: Theory and Applications

Domain and Range of the Relation
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 Discrete Mathematical Structures: Theory and Applications

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) Discrete Mathematical Structures: Theory and Applications

Relations Constructing New Relations from Existing Relations

Relations Example: Consider the relations R and S as given in Figure 3.7. The composition S ◦ R is given by Figure 3.8. Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

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 {A1, A2, A3, A4, A5} be a partition of A. Corresponding to this partition, a Venn diagram, can be drawn, Figure 3.13 Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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) Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Figure 3.24 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. Discrete Mathematical Structures: Theory and Applications

Figure 3.24 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 Discrete Mathematical Structures: Theory and Applications

Non-distributive Lattice Because a ∧ (b ∨ c ) = a ∧ 1 = a = 0 = 0 ∨ 0 = (a ∧ b) ∨ (a ∧ c ), this is not a distributive lattice Discrete Mathematical Structures: Theory and Applications

Complement Let D30 denote the set of all positive divisors of 30. Then D30 = {1, 2, 3, 5, 6, 10, 15, 30}. (D30,≤) is a poset where a ≤ b if and only if a divides b (≤ is the divisibility relation) D30,≤) is a poset where a ≤ b if and only if a divides b (≤ is the divisibility relation). Because 1 divides all elements of D30, it follows that 1 ≤ m, for all m ∈ D30. Therefore, 1 is the least element of this poset. Discrete Mathematical Structures: Theory and Applications

Complement Let a, b ∈ D30. 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 ∈ D30 and c ≤ a, c ≤ b. This shows that d is a lower bound of {a, b}. Let c ∈ D30 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}. Discrete Mathematical Structures: Theory and Applications

Complement Because all the positive divisors of a, b are also divisors of 30, d ∈ D30, so d = a ∧ b It can be shown that m ∈ D30 and m = a ∨ b Hence, (D30,≤) is a lattice with the least element integer 1 and the greatest element 30 Discrete Mathematical Structures: Theory and Applications

Complement For any element a ∈ D30, (30/a) ∈ D30 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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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) Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

CSE 2353 OUTLINE Functions Sets Logic Proof Techniques
Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra

Explore various properties of functions
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 Discrete Mathematical Structures: Theory and Applications

Functions Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

The image of g is Im(g) = {a, b, c , d} = 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 Discrete Mathematical Structures: Theory and Applications

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 is satisfied,but the second one is not. Therefore, h is not a function Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

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 Discrete Mathematical Structures: Theory and Applications

Functions Example 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 a1, a2 ∈ A and a1 = a2, then f(a1) = f(a2). 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. Discrete Mathematical Structures: Theory and Applications

Functions Let A = {1,2,3,4} and B = {a, b, c , d, e}
Example Let A = {1,2,3,4} and B = {a, b, c , d, e} f : 1 → a, 2 → a, 3 → a, → 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. Discrete Mathematical Structures: Theory and Applications

Functions Let A = {1,2,3,4} and B = {a, b, c , d, e}
f : 1 → a, 2 → b, 3 → d, → 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. Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

Special Functions and Cardinality of a Set

Sequences and Strings Discrete Mathematical Structures: Theory and Applications

m is the lower limit of the sum, n the upper limit of the sum, and ai the general term of the sum

m is the lower limit of the sum, n the upper limit of the sum, and ai the general term of the product. Discrete Mathematical Structures: Theory and Applications

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. Discrete Mathematical Structures: Theory and Applications

Binary Operations Discrete Mathematical Structures: Theory and Applications

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 (ii), the concatenation operation is associative. Hence, A+ becomes a semigroup with respect to the concatenation operation. Called a free semigroup generated by A. Discrete Mathematical Structures: Theory and Applications

Binary Operations Let A* denote the set of all words including empty word λ. By Theorem (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 Mathematical Structures: Theory and Applications

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