chapter 1 part 1 new - stud

5/25/2018 Chapter 1 Part 1 New - Stud

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DCS5028: DISCRETE STRUCTURE

CHAPTER 1

(PART 1)

Introduction to Logic

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LEARNING OBJECTIVES

Differentiate between propositions andcompound proposition.

Write the compound propositions.

Differentiate and write the PropositionalEquivalences.

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LOGI Logic

The discipline that deals with the methods of

reasoning.

Provides rules and techniques for determiningwhether a given argument is valid.

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LOGIC

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All cats have four legs.

I have four legs.

I am a cat.

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PROPOSITIONS Proposition

Is a declarative sentence that is either TRUEor

FALSE, but NOT BOTH

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EXAMPLE

Which of the following sentences is proposition?

Determine the truth value(true or false) of thesentence if it is a proposition.

1. Singapore is not an ASEAN Country.

2. Apple is a fruit.3. Just follow the step-by-step instructions, which will guide

you through the questionnaire.4. In the film, Captain Jack Sparrow (Johnny Depp) is joined

by Angelica (Lady Gaga) in his search for the Fountain ofYouth.

5. 6 + 12 * 5 = 3666. x + y = 3337. Cat can fly.8. What a beautiful evening!

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(P, FALSE)

(P, TRUE)

Not a Proposition

(P, FALSE)

(P, FALSE)

Not a Proposition

(P, FALSE)(P, TRUE/ FALSE)


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EXERCISE

Which of the following are propositions?

The earth is round.

2 + 3 = 5

Listen!

5- x =3

Are you sleeping?

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COMPOUND PROPOSITIONS

In logic,letters p, q, rdenote propositional

variables.

Example:

p : Today is a windy day.

q: It is cold.

Propositional variables can be combined by

logical connectives to obtain compound

proposition.DCS5028

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OMPOUND PROPOSITIONS1. Negation

2. Conjunction

2. Disjunction3. Exclusive Disjunction (exor)

5. Implication/Conditional

6. Biconditional

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The above words are known as connect ives

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Connectives Symbol Name

not ~ negationand conjunction

or disjunction

exclusive Or exclusive

disjunction(or)

ifthen. Implication

/conditional

if and only if biconditional 10DCS5028


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NEGATION

Let pbe a proposition.

The negationof p, denoted as ~p or p, read as not

p.

Can be written in English as this is not the case that

p.

The operator is a unary operator on propositions.

The truth value of the proposition p is defined by thetruth table

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NEGATION

Example 1:

Let q: Ali can swim.

Answer : q

Example 2:

Let r: Amy is a dance instructor.

Answer: r

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Ali could not swim.Ano ther answer: It is not the case that Ali can swim

Amy is not a dance instructor

Ano ther answer: It is not the case that Amy is a dance instructor

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EXERCISE

Give the negation of the following

statements:

a) p : 2+3>1

b) q: It is cold

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CONJUNCTION

Let p and q be propositions, The propositionof p and q, denoted by p q. The proposition that is TRUEwhen both p and q

are TRUEand is FALSE otherwise.

The proposition p q is called a conjunct ion of pand q.

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p q p q

T T TT F F

F T F

F F F


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CONJUNCTION

Example 3:Form the conjunction of p and q for the following.

Let p: Ali enjoys running. Let q: Ali likes hiking.Answer:

QUESTIONWhat is the truth value of compound proposition above,

if the p proposition is TRUEand q proposition is

FALSE?

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Ali likes running and hiking. OrAli enjoys running and hiking.

FALSE

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EXERCISE:

Form the conjunction of p and q for each of

the following and determine the truth value

for the compound proposition:

a) p: 2 < 3 q: -5 > -8

b) p: 2 + 5 7 q: 2


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DISJUNCTION

Let p and q be propositions. The disjunction of p or q, denoted by p q, is a

compound proposition means p or q.

Disjunction of p or q is the proposition that is FALSE

when p and q are both false and TRUE otherwise.

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p q p V q

T T TT F T

F T T

F F F


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DISJUNCTION

Example 4Let p: John is at the library.Let q: John is studying.

Answer:

QUESTION

What is the truth value of propositions in example

above, if the pproposition is falseand the qproposition

is true?

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John is at the library or John is studyingOr

John is either at the library or he is studying

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EXERCISE:

Form the disjunction of p and q for each of

the following and determine the truth value

for the compound proposition:

a) p: 2 < 3 q: 2 is a positive integer

b) p: 2 + 5 7 q: 2 is a positive integer

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EXCLUSIVE DISJUNCTION(EXOR)

Let p and q be propositions.

The exclusive or of p and q, is denoted by p q.

Exclusive OR of p q is the proposition that is

TRUE when EXACTLY ONE of the propositions(p,q)

is TRUE and is FALSE otherwise.

Example:

p: I left for Singapore on Monday

q: I left for Singapore on Wednesday

Exactly one of the two possibilities could haveoccurred. Both could not have occurred.

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EXCLUSIVE-OR

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p q p q

T T F

T F TF T T

F F F


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EXCLUSIVE DISJUNCTION(EXOR)

Example 5:Let p: Muthu likes to eat. q: Muthu likes to watch

movies.

Answer:

QUESTION

What is the truth value of propositions in exampleabove, if both p and qpropositions are false?

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Muthu likes to eat exclusive or watching movies

FALSE

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IMPLICATION

Let p and q be propositions.

The implication of p and q, denoted by pq, is a

compound proposition that means if p then q.

In this implication, p is called the hypothesis and q is

called the conclusion (or consequence).

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EXERCISE

Form the implication pq for each of the

following:

a) p: I am hungry q: I will eat

b) p: It is raining q: 3+ 5 = 8

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IMPLICATION

For an implication :

Theconverse of it is q p

The contrapositiveof it is q p

The proposition of p q is called the inverseof pq.

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Example 6

Let p: Muthu likes to eat. q: Muthu likes to watch movies.

1. pq:

2. Converse:

3. Contrapositive:

4. Inverse:

Answer:

1. If Muthu likes to eat, then he likes to watch movies.

2. If Muthu likes to watch movies, then he likes to eat.3. If Muthu doesnt like to watch movies, then he doesnt

like to eat

4. If Muthu doesnt like to eat, then he doesnt like to

watch movies

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IMPLICATION (CONTINUED)

What is the truth value for (1) to (4) in

Example 6, if pis TRUE and qis FALSE?

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JUST FOR FUN

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IF

THEN

1 1 11

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What is the truth value for this implication?

Answer: TF = F


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BICONDITIONAL

Let p and q be propositions. The biconditional of p if and only if q, denoted by p

q.

The Biconditional for p q is TRUE when p and q have

the same truth values, and is FALSE otherwise.

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BICONDITIONAL

Example 7Let p: Muthu likes to eat. q: Muthu likes to watch

movies.

Answer:

QUESTION

What is the truth value of propositions in example given,

if both pand qproposition are FALSE?

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Muthu likes to eat if and only if he likes to watch movies.

TRUE

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EXERCISE

Is the following biconditional a true

statement?

3 > 2 if and only if 0 < 3-2.

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REVIEW QUESTIONS

1. Which of these sentences are propositions? What are the

truth-values of those that are propositions?

2 * 4 = 8

Stand UP!!!.

Mr. Mat Aris is the Manager for CDP.

2. What is the negationfor each of the propositions below? PHP is an Open Source Software.

(24 * 12) / 3 = 36.

3. Given that proposition ris true, proposition sis trueandproposition mis false, determine whether each

proposition below is true or false.

r m

r (m s)31

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REVIEW QUESTIONS

4. Let p and q be the propositions Michael likes scuba

diving and Michael likes hiking. Express each of these

compound propositions as an English sentence. q

p q

p q

p

q5. State the inverse, converse and contrapositive of the

following proposition: If an object is a triangle then it is a

polygon.

6. Construct a truth table for each of these compoundpropositions.

p q

p (q r )

(p q) (pq)

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REVIEW QUESTIONS

7. Letpand qbe the propositions

p: Your car is out of gas.

q: You can't drive your car.

Write the following propositions using p and qand logical

connectives.

Your car is not out of gas.

You can't drive your car if it is out of gas. Your car is not out of gas if you can drive it.

If you can't drive your car then it is out of gas.

8. Given that proposition pis false, proposition qis true, and

proposition ris false, determine whether each propositionbelow is true or false.

p q

p q

(p q) (p r)33

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