UNISZA:TAF3023: DISCRETE MATHEMATICS PRESENTATION 1:FORESPEC GROUP

7/29/2019 UNISZA:TAF3023: DISCRETE MATHEMATICS PRESENTATION 1:FORESPEC GROUP

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DISCRETE MATHEMATICS

FORESPEC MEMBER:

Nor Farahana Zainul Hisham (032291)

Nurul Syazwani Kamarulzaman(032691)

Nor Shila Latif (032840)

Nor Farahin Rosli (032963)

Nor Irma Fariza Mohd Zamri (033053)

Sam Tau Siong (032261)Muhammad Husni Ideris (032500)


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CHAPTER 1

PROPOSITIONAL LOGIC

Definition : proposition and example

Definition propositional variable

Types of the true table

PROPOSITIONAL EQUIVALENCES Definition : Tautology, contradiction, contingency

Logical equivalences

PREDICATES AND QUANTIFIERS

Introduction of predicates Quantifies (definition and example)

Example of using quantifiers in reality


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PROPOSITIONAL LOGIC

What is

proposition?

Proposition or sometimes called statement.

A proposition is a declaration statement which is

either true or false, but not both simultaneously.


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PROPOSITIONAL LOGIC

This rose is red

Triangles have 4 vertices

9+2=11

6


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Logic Connectives


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Negation

While negation is an operation which involves only a singleproposition, logical connectives are used to link pairs of

propositions.

We can summarize this in table. Ifp symbolizes a

proposition ~p symbolizes the negation ofp.


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5 commonly used logical

connectives:

1. Conjunction

If p and q are two propositions p ^ q (or p.q ) symbolizes the

conjunction of p and q. p q is true when both p and q are true.

Otherwise the conjunction is false. For example:

p : The sun is shining.

q : Cow eat grass.

p ^ q: The sun is shining and cow eat grass.
http://3.bp.blogspot.com/-0ZXtAdn0qIY/USdB9JP3CQI/AAAAAAAAABo/mk29ojiZ4yk/s1600/n.pnghttp://3.bp.blogspot.com/-0ZXtAdn0qIY/USdB9JP3CQI/AAAAAAAAABo/mk29ojiZ4yk/s1600/n.png


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connectives:

2. Inclusive Disjunction

Given the two propositions p and q, p V q symbolizes

the Inclusive Disjunction of p and q. This

compound propositions is true when either or both of its

components are true and false otherwise. Thus the true tablefor p V q is given by:


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connectives:

3. Exclusive Disjunction

The exclusive disjunction ofp and q is symbolizes by p V q. This

compound propositions is true when exactly one of the

compound is true. Example :

"Tomorrow i will go swimming or play golf"seems to suggest

that will not do both and therefore points to exclusive

disjunction.

The truth table forp V q is given by:


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connectives:

4. Conditional Propositions

The conditional connective (implication) is symbolized by

().The linguistic expression of a conditional proposition is

normally accepted as utilizing ' if...then..' as in the following

example:

p : I eat breakfast.q : I don't eat lunch

pq :If eat breakfast then i don't eat lunch.

Notice that the proposition 'if p then q ' is false only whenp is true

and q is false. A true statement cannot imply a false one. Ifp is false, thecompound proposition is true no matter what the truth value q .


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connectives:

5. Biconditional Propositions

The biconditional connective is symbolized by (),and

expressed by 'if and only if...then...'.Using the previous example:

p : I eat breakfast.

q : I don't eat lunch.

pq :If eat breakfast if and only if i don't eat lunch.

Note that for pq tobe true, p and q must both have same truthvalues. Both must be true or both must be false.


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PROPOSITIONAL VARIABLEWhat is

propositional

variable?

variable which can either be true or false.

we use letters p,q,r,to denote propositional

variables.


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TYPE OF TRUTH TABLE

1. Constant true

in example: it is always return

true

P value returned

T TF T

2. Identity


the value of p

P value returned

T TF F

3. Negation

P value returned

T F

F T

4. Constant false


false

P value returned

T F

F F


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TAUTOLOGY

Tautology is ..

A compound proposition or statement form that is true and

never false for every assignment of truth values to its

components

Example: q q / q or not q

q q q qTrue False True

False True True


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CONTRADICTION

A compound proposition or statement form that is false

and never true for every assignment of truth values to

its components.

q q / q and not q

Example:

q q q qTrue False False

False True False


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A compound proposition or statement that isneither tautology nor contradiction.

CONTIGENCY

Example:

(p q) r / p and q or not r

p q r p q r (p q) r

True True True True False True

True True False True True True

True False True False False False

True False False False True True

False True True False False False

False True False False True True

False False True False False False

False False False True True True


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LOGICAL EQUIVALANCE

The proposition p and q are called logically equivalentif p q is a tautology.

The notation of p q denotes that p and q are logically

equivalent.

The symbol is not logical connective since p q is

not a compound proposition, but rather is the

statement that p q is a tautology.

The symbolis sometimes used instead of todenotes logical equivalence.


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How to determine the equivalence?

One way to determine whether two

propositions are equivalent is to use a truth

table.

The propositions p and q are equivalent if and

only if the columns giving their truth values

agree.


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List of Equivalences

pT p; pF p Identity Laws

pT T; pF F Domination Laws

pp p; pp p Idempotent Laws

(p) p Double Negation Law

pq qp; pq qp Commutative Laws

pq qp; pq qp Commutative Laws

(pq) r p (qr)

(pq) r p (qr) Associative Laws


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p(qr) (pq)(pr)

p(qr) (pq)(pr)

Distribution Laws

(pq)(p q)

(pq)(p q)

De Morgans Laws

p p T

p p F

(pq) (p q)

Or Tautology

And Contradiction

Implication Equivalance

pq(pq) (qp)

Biconditional

Equivalence


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Truth Tables for (p q) and (p q)

p q P q (p q) p q p qT T T F F F F

T F T F F T F

F T T F T F F

F F F T T T T


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Truth Tables for (p q) and p q

p q p (p q) p qT T F T T

T F F F F

F T T T TF F T T T


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Logical Equivalences involving Implications

p q p q

p q q p

p

q pqp q (pq)

p q p q

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

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

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


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Logical Equivalence involving Biconditionals

p q (p q)(q p)

p q p q

p q (p q) (p q)

p q p q


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PREDICATESPredicates is study of

propositional logic.

For example:

I have more than 5 candy

The statement is I.

While more than 5 candyis predicates.

With the instruction above, we know that I

have more than 5 candy,

it's cannot be 4 or a3,because it cannot be less.


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QUANTIFIER

Here is a (true) statement about real numbers:

Every real number is either rational or irrational

I could try to translate the statement as follows: let

b = x real number

c = x is irrational

d = x is rational

The statement can be expressed as the implication b (c V d)

but as you can see that Im cheating in making my translation:

x is a real number

Which is not uniquely specified object x. Which different from

is a real number

Which talk a specific real number


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QUANTIFIER

So that I can useQuantifiers to translate statement like those so as tocapture this meaning. Mathematicians use two quantifier:

, the universal quantifier, which is read "for all", "for every", or "for each".

, the existential quantifier, which is read "there is" or "there exists".


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QUANTIFIER

Eg Let mean x likes pizza. Then :

means Everyone likes pizza

means Someone likes pizza

If Someone likes pizza is true

It maytrue that everyone like pizza(by assuming the set of people is nonempty)It mustbe true that Someone likes pizza

means Not everyone likes pizza.

means No one likes pizza

Again it same as above.


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Example Of Using Of Quantifiers In Reality

Program Verification

-proving or disproving the correctnessSystem Specification

-explain on function and description of the system

Logic Programming-execute by user

UNISZA:TAF3023: DISCRETE MATHEMATICS PRESENTATION 1:FORESPEC GROUP

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