DISCRETE COMPUTATIONAL STRUCTURES

CSE 2353

Fall 2010

Most slides modified from Discrete Mathematical Structures: Theory and Applications

by D.S. Malik and M.K. Sen

 

CSE 2353 OUTLINE

PART I1. Sets 2. LogicPART II3. Proof Techniques4. Relations 5. FunctionsPART III6. Number Theory7. Boolean Algebra

CSE 2353 OUTLINEPART I1. Sets 2. LogicPART II

3.Proof Techniques4. Relations 5. Functions

PART III6. Number Theory7. Boolean Algebra

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Proof Techniques

Direct Proof or Proof by Direct MethodProof of those theorems that can be

expressed in the form ∀x (P(x) → Q(x)), D is the domain of discourse

Select a particular, but arbitrarily chosen, member a of the domain D

Show that the statement P(a) → Q(a) is true. (Assume that P(a) is true

Show that Q(a) is trueBy the rule of Universal Generalization (UG), ∀x (P(x) → Q(x)) is true

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Proof Techniques

Indirect Proof

The implication p → q is equivalent to the implication (∼q → ∼p)

Therefore, in order to show that p → q is true, one can also show that the implication (∼q → ∼p) is true

To show that (∼q → ∼p) is true, assume that the negation of q is true and prove that the negation of p is true

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Proof Techniques

Proof by Contradiction Assume that the conclusion is not true and then

arrive at a contradictionExample: Prove that there are infinitely many prime

numbersProof:

Assume there are not infinitely many prime numbers, therefore they are listable, i.e. p1,p2,…,pn

Consider the number q = p1p2…pn+1. q is not divisible by any of the listed primes

Therefore, q is a prime. However, it was not listed.Contradiction! Therefore, there are infinitely many

primes.

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Proof Techniques

Proof of Biimplications To prove a theorem of the form ∀x (P(x) ↔

Q(x )), where D is the domain of the discourse, consider an arbitrary but fixed element a from D. For this a, prove that the biimplication P(a) ↔ Q(a) is true

The biimplication p ↔ q is equivalent to (p → q) ∧ (q → p)

Prove that the implications p → q and q → p are trueAssume that p is true and show that q is trueAssume that q is true and show that p is true

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Proof Techniques

Proof of Equivalent Statements

Consider the theorem that says that statements p,q and r are equivalent

Show that p → q, q → r and r → p Assume p and prove q. Then assume q

and prove r Finally, assume r and prove p

What other methods are possible?

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Other Proof Techniques

Vacuous

Trivial

Contrapositive

Counter Example

Divide into Cases

Constructive

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Proof Basics

You can not prove by example

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Proofs in Computer Science

Proof of program correctness

Proofs are used to verify approaches

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Mathematical Induction

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Mathematical Induction

Proof of a mathematical statement by the principle of mathematical induction consists of three steps:

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Mathematical InductionAssume that when a domino is knocked over, the next domino

is knocked over by itShow that if the first domino is knocked over, then all the

dominoes will be knocked over

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Mathematical Induction

Let P(n) denote the statement that then nth domino is knocked over

Show that P(1) is trueAssume some P(k) is true, i.e. the kth domino is

knocked over for some

Prove that P(k+1) is true, i.e.

1k

)1()( kPkP

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Mathematical Induction

Assume that when a staircase is climbed, the next staircase is also climbed

Show that if the first staircase is climbed then all staircases can be climbed

Let P(n) denote the statement that then nth staircase is climbed

It is given that the first staircase is climbed, so P(1) is true

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Mathematical Induction

Suppose some P(k) is true, i.e. the kth staircase is climbed for some

By the assumption, because the kth staircase was climbed, the k+1st staircase was climbed

Therefore, P(k) is true, so

1k

)1()( kPkP

CSE 2353 OUTLINEPART I1. Sets 2. LogicPART II

3. Proof Techniques4.Relations

5. FunctionsPART III

6. Number Theory7. Boolean Algebra

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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

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Relations

Relations are a natural way to associate objects of various sets

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Relations

R can be described in

Roster form

Set-builder form

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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

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Relation Arrow Diagram

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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

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Relation Directed Graph

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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

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Inverse of Relations

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Relations

Constructing New Relations from Existing Relations

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Composition of Relations

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Properties of Relations

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Relations

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Relations

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Equivalence Classes

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Partially Ordered Sets

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Partially Ordered Sets

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Partially Ordered Sets

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Partially Ordered Sets

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Partially Ordered Sets

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.

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Partially Ordered Sets

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Digraph vs. Hasse Diagram

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Minimal and Maximal Elements

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Partially Ordered Sets

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Partially Ordered Sets

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Application: Relational Database

A database is a shared and integrated computer structure that storesEnd-user data; i.e., raw facts that are of interest

to the end user;Metadata, i.e., data about data through which

data are integratedA 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

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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

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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

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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)

A relational database management system provides a standard language, called structured query language (SQL)

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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

PART I1. Sets 2. LogicPART II

3. Proof Techniques4. Relations

5.FunctionsPART III

6. Number Theory7. Boolean Algebra

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Learning Objectives

Learn about functions

Explore various properties of functions

Learn about binary operations

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Functions

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Functions

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Functions

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Functions

Every function is a relationTherefore, 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.

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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: 1) 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

2) 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

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Functions

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Functions

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Functions

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Special Functions and Cardinality of a Set

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Special Functions and Cardinality of a Set

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Special Functions and Cardinality of a Set

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Special Functions and Cardinality of a Set

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Special Functions and Cardinality of a Set

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Special Functions and Cardinality of a Set

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Mathematical Systems