Chapter 5- Logic

8/11/2019 Chapter 5- Logic

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Chapter 5 Logic

CSNB 143

Discrete Mathematical Structures


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OBJECTIVES

Student should be able to know what is it meansby statement.

Students should be able to identify its

connectives and compound statements.

Students should be able to use the Truth Table

without difficulties.


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Logic

Statement or proposition is a declarative sentence withthe value of true or false but not both.

Ex 1: Which one is a statement?

The world is round. 2 + 3 = 5

Have you taken your lunch?

3 - x = 5

The temperature on the surface of Mars is 800F. Tomorrow is a bright day.

Read this!


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Statement usually will be replaced by variablessuch as

p, q, r or s.

Ex 2:

p: The sun will shine today.

q: It is a cold weather.

Statements can be combined by logicalconnectives to obtain compound statements.

Ex 3:

AND (p andq): The sun will shine today andit isa cold weather.


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Connectives ANDis what we called conjunctionfor p and q, written p q. The compound

statement is true if both statements are true.

Connectives ORis what we called disjunction

for p and q, written p q. The compound

statement is false if both statements are false.

To prove the value of any statement (or

compound statements), we need to use the Truth

Table.


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P Q P Q P Q

T T T T

T F F T

F T F T

F F F F


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Negationfor any statement p is notp. written as~p or p. The Truth Table for negation is:

Ex 4: Find the value of (~pq) pusing TruthTable.

P P

T F

F T


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P Q P P Q ( P Q) P

T T F F T

T F F F TF T T T T

F F T F F


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Conditional Statements If p and q are statements, the compound

statement if p then q, denoted by p q is calleda conditional statementor implication.

Statement p is called the antecedentor

hypothesis; and statement q is calledconsequentor conclusion. The connective if thenis denoted by the symbol .

Ex 5:

a) p : I am hungry q : I will eat b) p : It is cold q : 3 + 5 = 8

The implication would be: a) If I am hungry,then I will eat.

b) If it is cold, then 3 + 5 =


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Take note that, in our daily lives, Ex 5 b) has noconnection between statements p and q, that is,statement p has no effect on statement q.However, in logic, this is acceptable. It showsthat, in logic, we use conditional statements in a

more general sense. Its Truth Table is as below:

To understand, use: p = It is raining q = I used umbrella

P Q P Q

T T T

T F F

F T T

F F T


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Another meaning that use the symbol includes:

if p, then q

p implies q

if p, q p only if q

p is sufficient for q

q if p

q is necessary for p


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If p q is an implications, then the converseof itis the implication q p, and the contrapositive

of it is ~q ~p.

Ex 6:

p = It is raining q = I get wet

Get its converse and contrapositive.


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If p and q are statements, compound statement pif and only if q, denoted by p q, is called an

equivalenceor biconditional.

Its Truth Table is as below:

To understand, use: p = It is raining q = I used

umbrella

Notis that p q is True in two conditions: both p and q are True,or both and are false.

P Q P Q

T T T

T F F

F T FF F T


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Another meaning that use the symbol includes:

p is necessary and sufficient for q

if p, then q, and conversely

In general, compound statement may contain few

parts in which each one of it is yet a statement

too.

Ex 7: Find the truth value for the statement(p q) (~q ~p)


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P Q PQ

(A)

Q P QP

(B)

(A)(B)

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

A statement that is true for all possible values of

its propositional variables is called a tautology.


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A statement that is always false for all possiblevalues of its propositional variables is called a

contradiction.

A statement that can be either true or false,

depending on the truth values of its propositional

variables is called a contingency


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Logically Equivalent

Two statements p and q are said to be logically

equivalent if p q is a tautology.

Ex 8: Show that statements p qand (~p) q

are logically equivalent.


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Quantifier

Quantifier is used to define about all elements

that have something in common.

Such as in set, one way of writing it is {x | P(x)}

where P(x) is called predicate orpropositional

function, in which each choice of x will produces

a proposition P(x) that is either true or false.


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There are two types of quantifier being used:

a) Universal Quantification () of a predicate P(x)

is the statement

For all values of x, P(x) is true

In other words:

for every x

every x

for any x


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b) Existential Quantification () of a predicate P(x)is the statement

There exists a value of x for which P(x) is

true

In other words:

there is an x

there is some x there exists an x

there is at least one x


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Theorem 1

Operations on statements are:

Commutative

p q q p

p

qq

pAssociative

p (q r) (p q) r

p (q r) (p q) r

Distributive

p (q r) (p q) (p r)

p (q r) (p q) (p r)


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Idempotent

p p p

p p p

Negations

~(~p) p

~(p q) (~p) (~q)

~(p q) (~p) (~q)


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Theorem 2

Operations on implications are:

(p q) ((~p) q)

(p q) ((~q) ~p)

(p

q)((p

q)

(q

p))

~(p q) (p ~q)

~(p q) ((p ~q) (q ~p))


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Theorem 3

Operations on quantifier are:

~(x P(x)) x ~P(x)

~(x ~P(x))

x P(x)

x (P(x) Q(x)) x P(x) x Q(x)

x P(x) x Q(x) x (P(x) Q(x))

Anyone can achieve Q if you fulfill P

Even though you satisfy P, only someone can achieve Q


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((x P(x)) (x Q(x))) x (P(x) Q(x))

tautology

x (P(x) Q(x)) x P(x) x Q(x)tautology


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Chapter 5- Logic