BIT 1113Discrete

Mathematics

Dr. Mohd.Najib B. Mohd.Salleh

Department of Information System

Faculty of Information Technology & Multimedia

Course Material

Textbook:

(1)Johnsonbaugh R,(2005). Discrete Mathematics. Fifth Edition. Singapore: Prentice Hall, Inc. Other Reference:

Kolman, B., Busby, R., Ross, S. (1996). Discrete Mathematical Structures. New Jersey: Prentice Hall, Inc.

D.S Malik and M.K.Sen. (2004).Discrete Mathematical Structures : Theory & Application United State: Thomson Course Technology.

James L. H. (2002). Discrete Structures, Logic, and Computability. Second Edition. John and Bartlett Pub. Co.

Course Assessment

Test 2x 40%

Assignment ?x 10%

quizzes 5x 10%

Final Exam 40%

• Is a language of science

Mathematics Mathematics Words of wisdomWords of wisdom

Mathematics

Other disciplines

Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone – Albert Einstein

For the things of this world cannot be made known without a knowledge of mathematics- Roger Bacon

- Today’s computer systems are more complex, and more rapidly evolving. Thus, need for tools and techniques that assist in understanding the behavior of the systems.

- This will answer the questions such as cost and performance of the systems.

- 2 ways : experimental and modeling

Mathematics for ITMathematics for IT

What is discrete mathematics?

logic, sets, functions, relations, etc

Logic: artificial intelligence (AI), database, circuit design

Number theory: cryptography, coding theory

Counting: probability, analysis of algorithm

Graph theory: computer network, data structures

Introduction to Discrete Mathematics

Mathematics in Computer Science

Design efficient computer systems.

•How did Google manage to build a fast search engine?

•What is the foundation of internet security?

algorithms, data structures, database,

parallel computing, distributed systems,

cryptography, computer networks…

Logic, number theory, counting, graph theory…

Algebra in Programming, Database Algebra in Programming, Database and Data Miningand Data Mining

Algebra:

Set, Function, Relation, and Number Theory, Graph Theory, etc.

Why learn Algebra?

It’s a basis for Mathematics which is tool for reasoning.

Algebra in Programming, Database Algebra in Programming, Database and Data Miningand Data Mining

Why is Algebra important for Computer Science? Useful tool for formalizing and reasoning about computation and the objects of computation.

Algebra, especially set theory is indivisible from Logic where Computer Science has its roots. It has been and is likely to continue to be a source of fundamental ideas in Computer Science from theory to practice.

Given a set of items I={I1,I2,…,Im} and a database of

transactions D = {t1,t2,…tn} where ti={Ii1,Ii2,…,Iik } and Ii1 I, an

association rule is an implication of the form X=> Y where X,Y I are sets of items called itemsets and X Y =

The support (s) for an AR X=>Y is the percentage of

transactions in the database that contain X U Y.

The confidence or strength (α) for an AR X=>Y is the ratio of the number of transactions that contain X U Y to the number of transactions that contain X.

Data Mining : support and confidence

FUNDAMENTALS OF

MATHEMATICAL LOGIC

Logic and Proofs

Logic: propositional logic, first order logic

Proof: induction, contradiction

How do computers think?

Artificial intelligence, database, circuit, algorithms

Proposition

Statements that are not propositions include questions and commands.

A declarative sentence or any meaningful statement that is either True or False, but not both.

use lowercase letters, such as p, q, r, · · · , to represent propositions notation. p : 2 + 2 = 3

P defined as the proposition 2+2 = 3.

The truth value of a proposition :

true, denoted by T, if it is a true statement false, denoted by F, if it is a false statement.

Proposition

Example 1.1

Which of the following are propositions? Give the truth value of the propositions.

a. 5 - 11 = 7.

b. Dato’ Prof Ismail Bakar is a vice-chancellor of UTHM.

c. How do you do?

d. Look at the weather!

Example 1.2

Which of the following are propositions? Give the truth value of the propositions.

a.You and me.

b.2 is even number

c. Kuala Lumpur is the capital city of Malaysia

d. How are you?

propositional variables

New propositions can be constructed from existing propositions by using symbolic connectives or logical operators.

AND:: OR::

Let p and q be propositions.

The conjunction of p and q,

denoted p Λ q,

is the proposition: p and q.

This proposition is defined to be true only when both p and q are true and it is false

otherwise.

Conjunction

Let p and q be propositions.

The disjunction of p and q,

denoted p V q,

is the proposition: p or q.

The ’or’ is used in an inclusive way. This proposition is false only when both p and q are

false, otherwise it is true.

Disjunction

Example 1.3

Assume that

p : 5 < 9

q : 9 < 7

Construct the propositions p Λ q and p V q.

Solution.The conjunction of the propositions p and q is the

proposition p Λ q : 5 < 9 and 9 < 7

The disjunction of the propositions p and q is the proposition

p V q : 5 < 9 or 9 < 7

Example 1.4

Consider the following propositions

p : It is a monkey

q : It is a banana.

Construct the propositions p Λ q and p V q.

The disjunction of the propositions p and q is the proposition p V q : It is a monkey or it is a banana

Solution.The conjunction of the propositions p and q is the proposition p Λ q : It is a monkey and it is a banana

Truth Table

A truth table displays the relationships between the truth values of propositions

F

F

F

T

P Q

FF

TF

FT

TT

QP

F

T

T

T

P Q

FF

TF

FT

TT

QP

Exclusive-Or

Let p and q be two propositions.

The exclusive or of p and q, denoted p q, is the proposition that is true when exactly one of p and q is true and is false otherwise.

The truth table of the exclusive ’or’ is displayed below

p q p qT T F

T F T

F T T

F F F

Example 1.5

a. Construct a truth table for (p q) r

Solution.

p q r p q (p q) r

T T T F T

T T F F F

T F T T F

T F F T T

F T T T F

F T F T T

F F T F T

F F F F F

Example 1.5

b. Construct a truth table for (p p)

Solution. p p

T F

F F

The final operation on a proposition p that we discuss is the negation of p.

The negation of p, denoted ~ p, is the proposition not p.

The truth table of ~ p is displayed below

p ~ p

T F

F T

Negation (Logical Operator)

Example 1.6

Find the negation of the proposition

p : -5 < x = 0

Solution

The negation of p is the proposition

~ p : x > 0 or x = -5

Example 1.7

Consider the following propositions:

p: UTHM is located in Parit Raja.q: 2 + 5 = 7.r: There is no pollution in Parit Raja.

Question :

Construct the truth table of [~ (p Λ q)] V r.

solution

p q r p Λ q ~ (p Λ q) [~ (p Λ q)] V r

T T T T F T

T T F T F F

T F T F T T

T F F F T T

F T T F T T

F T F F T T

F F T F T T

F F F F T T

p q r p Λ q ~ (p Λ q) [~ (p Λ q)] V r

Tautology

A compound proposition is called a tautology if it is always true, regardless of the truth values of the basic propositions which comprise it.

Example 1.8

a. Construct the truth table of the proposition

(p Λq) V (~ p Λ ~ q).

Determine if this proposition is a tautology.

p q ~ p ~ q ~ (p V ~ q) (p Λ q) (p Λ q) V (~ p Λ ~ q)

T T F F F T T

T F F T T F T

F T T F T F T

F F T T T F T

Example 1.8

Determine if this proposition is a tautology.

b. Show that p V ~ p is a tautology.

Equivalent

Two propositions are equivalent if they have exactly the same truth values under all circumstances.

We write p = q.

Example 1.9

a. Show that ~ (p V q) =~ p Λ ~ qb. Show that ~ (p Λ q) =~ p V ~ qc. Show that ~ (~ p) = p. a. and b. are known as DeMorgan’s laws.

(Do it your self ! )

Conditional and

Biconditional Propositions

Let p and q be propositions.

Implication

The connective → is called the conditional connective.

p is called the hypothesis and q is called the conclusion.

The implication p→q is the proposition that is false only when p is true and q is

false; otherwise it is true.

Example 2.1

Construct the truth table of the implication p→ q.

p q p → q

T T T

T F F

F T T

F F T

Solution: The truth table is

Solution:

~ p V q.

p q ~ p p → q ~ p V q

T T F T T

T F F F F

F T T T T

F F T T T

Example 2.2

Show that p→ q

In terms of words the proposition p → q also reads:

(d) q is a necessary condition for p.

(b) p implies q.

(c) p is a sufficient condition for q.

(a)if p then q.

(e) p only if q

Example 2.3

Use the if-then form to rewrite the statement ”I become sleepy if I eat ‘nasi lemak’ in the morning.”

,

Solution:

If I eat ‘nasi lemak’ in the morning then I become sleepy.

In propositional functions that involve the connectives ~, , and the order of operations is that ~ is performed first and is performed last.

Thank you

As a conclusion:

Understand the concept of fundamentals of math logic,

What is proposition and other terminology in logic,

Construct the compound propositions, and

display the truth table.