Lecturer: Badrul Hisham Lecturer: Badrul Hisham AbdullahAbdullah

Kulliyyah MuamalatKulliyyah Muamalat

Mathematics IMathematics IMTD1013MTD1013

SUBJECT DESIGN

KOLEJ UNIVERSITI INSANIAH (KUIN)KULLIYYAH MUAMALAT

Semester II Session 2009/2010

Subject: Mathematics I

Course Code:

Lecturer:

Office:

Tel:

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Required Text Book:

MTD1013

En. Badrul Hisham Abdullah

Level 3 of Kulliyyah Muamalat

019-4476860

badrul_abdullah09@yahoo.com

Rosen K. H., Discrete Mathematics & Its Applications. (Sixth Edition), Mc Graw Hill, 2009.

WHAT IS DISCRETE MATHEMATICS?

Discrete Mathematics describes the processes that consist of a sequence of individual steps.

CHAPTER 2

LOGIC

What is about this chapter?

This chapter will introduce to us some of the basic tools of discrete mathematics. In the end of this chapter, it will give us

the background needed to begin the exploration of mathematics structures.

2.0 Logic and Methods of Proof

Definition

A statement or proposition is a declarative sentence that either true or false, but not both.

2.0.1 Logic

Example The earth is round. (Proposition - true)

a) 2 + 3 = 5. (Proposition - true) b) Where do you come from? (Question) c) Please seat down. (Command)

Notation In logic, the letters p, q, r, denote propositional variables; that is, variables that can be replaced by statements. Thus we can write p: The sun is shining today, q: It is cold. Statements or propositions can be combined by logical connectives to obtain compound statements.

Definition Let p and q be two statements.

a) The conjunction of p and q is the compound statement “p and q”, denoted by qp . b) The disjunction of p and q is the compound statement “p or q”, denoted by qp .

Definition If p is a statement, the negation of p is the statement not p, denoted by ~ p. The truth values of qp , qp and ~ p are given by truth tables below.

p q qp qp ~ p T T T T F T F F T F F T F T T F F F F T

Definition If p and q are statements, the compound statement “if p then q”, is called a conditional statement, or implication, denoted by p q . The statement p is called the hypothesis, and the statement q is called the conclusion. The truth values of p q are given in the following table.

p q p q T T T T F F F T T F F T

Definition If p and q are statements, the compound statement “p if and only if q”, is called an equivalence, or biconditional, denoted by p q . The truth values of p q are given in the following table.

p q p q T T T T F F F T F F F T

Definition If p q is a conditional statement, then the converse of p q is the conditional statement q p , and the contrapositive of p q is the conditional statement ~ ~q p . The compound statement p q and ~ ~q p are logically equivalent, and the truth values are given by the following table.

p q p q ~ q ~ p ~ ~q p ( ) (~ ~ )p q q p T T T F F T T T F F T F F T F T T F T T T F F T T T T T

Methods of Proof

Consider the conditional statement p q . If the statement is true, then we say that q logically

follows from p. Suppose that the conditional statement of the form 1 2( )np p p q is

true, then we say that q logically follows from 1 2, , , np p p .

This means if we know that 1p is true, 2p is true, , and np is true, then we know q is true.

The ip ’s are called the hypotheses, and q is called the conclusion.

To prove the theorem, we have four methods of proof:

A. Direct method B. Indirect method C. Contradiction method D. Mathematical Induction

A.Direct Method

Consider the conditional statement p q . We want to show that q will be true if all the ip are

true. Example Prove p q for the following p and q: p: number can be divided by 6. q: number can be divided by 3. Solution Suppose that x is a number that can be divided by 6. So x/6 = k, k is an integer x = 6k . But 6 = 2.3. Hence x = 2.3 k = 3.2 k = 3(2k) = 3l, l = 2k integer Therefore x is a number that can be divided by 3.

B. Indirect Method

This proof technique is based on ( ) ((~ ) (~ ))p q q p . Example As Example 1.36. ~ p: number cannot be divided by 6. ~ q: number cannot be divided by 3. Solution Suppose that x is a number that cannot be divided by 3. So x k.3, for any integer k x (2k).3, 2k integer x (k2).3, x k.6 . Therefore x is a number that cannot be divided by 6.

C. Contradiction Method

This proof technique is based on (( ) (~ )) (~ )p q q p . Assume that the hypothesis is true and the conclusion is false. Then we will get a contradiction somewhere. Example Proof that if x x x , then 0x . Solution Assume that x x x , and 0x . Now x x x xx 2 12 )0( x , a contradiction. Therefore the statement if x x x , then 0x is true.

D. Principle of Mathematical Induction

Objective: To proof that the statement P(n) is true for each integer 0nn .

Steps

Prove that the statement is true for 0nn .

Assume that the statement is true for n = k. Prove that the statement is true for 1kn . Conclusion: (PMI) The statement is true for every integer 0nn .

Example Use mathematical induction to show that

2)12(531 nn , (*) for any positive integers n.

Solution Let P(n): 2)12(531 nn . Prove that P(n) is true for n = 1, that is P(n = 1) is true.

Now P(n = 1) is 211 , which is clearly true. Let P(n = k) is true, that is

2)12(531 kk . (**) We want to prove that P(n = k+1) is true, that is

2)1(1)1(2531 kk . (***) Now left-hand side (***) = 1)1(2531 k = 1)1(2)1(2531 kk

= 1)1(22 kk (using (**))

= 122 kk = 2)1( k = right-hand side (***) . Therefore the statement (***) is true. By the PMI, the statement (*) is true for each positive integer n.

Chapter 1 Fundamentals BCT 1033 (Discrete Mathematics) Lecturer: En. Badrul Hisham Abdullah

Example By using mathematical induction, show that

1! 2 , 1,2,3,nn n (*) Solution Let P(n): 1! 2 , 1,2,3,nn n . Prove that P(n) is true for n = 1, that is P(n = 1) is true. Now

P(n = 1) is 12!1 0 , which is clearly true. Let P(n = k) is true, that is

12! kk . (**) We want to prove that P(n = k+1) is true, that is

kk 2)!1( . (***) Now left-hand side (***) = )!1( k

= 12)1(!)1( kkkk = 11 22 kkk But

)1(22.22222 11111 kk kkkkkk .

Thus kk 2)!1( = right-hand side (***). Therefore the statement (**) is true. By the PMI, then the statement (*) is true for every positive integers n.