1-disc math spring 2014-1

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Welcome to

2013-2014 SpringDiscrete Mathematics

Week 1: The Foundations - Logic1

eref Naci Engin, PhD EE

Yildiz Technical University

Faculty of Electrical & Electronics

Instructor: eref Naci Engin

Office:

Office Hours:

Phone:

E-Mail:

Website:



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Textbook for 0122232 Discrete Mathematics :



Evaluation

up to 20% (bonus)

2 x 30%

40%


Week 1: The Foundations - Logic

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Grading

5

For the exams and

course evaluations the

following scheme will be

used to convert marks

into letter grades:

2013-2014 Spring Discrete Mathematics - Week 1: The Foundations - Logic

Letter Grade 100

AA 4.00 90-100

BA 3.50 80-89

BB 3.00 70-79

BB 3.00 60-69

BB 3.00 53-59

CB 2.50 48-52

CC 2.00 40-47

DC 1.50 30-39

DC 1.50 0-29

DD 1.00 No attendance

FF 0.00

F0 0.00 50: DD

Attendance: min. 70% (FF)

Warning:

Why Care about Discrete Math?

structure (circuits) operations (execution of algorithms)



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Syllabus

The order of the topics may change



Propositional Logic

propositions

truefalse

truth valueT F

1 0



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The Statement/Proposition

9

Is this a statement? yes

Is this a proposition? yes

What is the truth value

of the proposition? true (T)




10




of the proposition? False (F)



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11


Is this a proposition? no

Its truth value depends on the value of y, but

this value is not specified.

We call this type of statement a propositional

function or open sentence.




12




of the proposition? false



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Is this a statement? no

Is this a proposition? no

Only statements can be

propositions.

Its a request.



Examples

Propositions 1 and 3 are true, whereas 2 and 4 are false.

Sentences 1 and 2 are not propositions because they are not declarative sentences.

Sentences 3 and 4 are not propositions because they are neither true nor false.

Spring, 2014Discrete Mathematics

Week 1: The Foundations: Logic and Proofs14

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Conditional Statements / Propositions

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of the proposition? probably false




16




of the proposition? true

because its truth value does not depend on

specific values of x and y.



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Combining Propositions

compound proposition

p q r slogical operators



Logical Operators (Connectives)

(NOT)

(AND)

(OR)

(XOR)

(if then)

(if and only if)



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Negation (NOT)

19

P P

true false

false true



Examples



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Examples

Janes smartphone has at least 32GB of memory

It is not the case that Janes smartphone has at least 32GB of memory.

Janes smartphone does not have at least 32GB of memory

Janes smartphone has less than 32GB of memory.



Conjunction (AND)

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P Q PQ

true true true

true false false

false true false

false false false



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Disjunction (OR)

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P Q PQ

true true true

true false true

false true true

false false false



Exclusive Or (XOR)

24

P Q PQ

true true false

true false true

false true true

false false false



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Conditional Statements:Implication (if - then)

25

P Q PQ

true true true

true false false

false true true

false false true



Implication (if - then), p q



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Example -1

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P Q PQ

T T T

T F F

F T T

F F T



Example -2

28

P Q PQ

T T T

T F F

F T T

F F T



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Clearing up the two confusions

29

P Q PQ

T T T

T F F

F T T

F F T

2013-2014 Spring Discrete Mathematics Week 1: The Foundations - Logic

p only if q (for pq):

q unless p (for pq):

Example -3

30

P Q PQ

T T T

T F F

F T T

F F T



Solution:

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Biconditional (if and only if)

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P Q PQ

true true true

true false false

false true false

false false true



Biconditional (if and only if)

p is necessary and sufcient for qif p then q, and converselyp iff q



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Example for if and only if statements



P Q PQ

T T T

T F F

F T F

F F T

This statement is true if p and q are either both true or both false, that is,

if you buy a ticket and can take the ight or

if you do not buy a ticket and you cannot take the ight.

It is false when p and q have opposite truth values, that is,

when you do not buy a ticket, but you can take the ight (such as when you get a free trip) and

when you buy a ticket but you cannot take the ight (such as when the airline bumps you).

Statements and Operators

34

P Q P Q (P)(Q)

true true false false false

true false false true true

false true true false true

false false true true true



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Statements and Operations

35

P Q PQ (PQ) (P)(Q)

true true true false false

true false false true true

false true false true true

false false false true true



Equivalent Statements

logically equivalent

36

P Q (PQ) (P)(Q) (PQ)(P)(Q)

true true false false true

true false true true true

false true true true true




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Tautologies and Contradictions

Examples:

R(R)

(PQ)(P)(Q)



Tautologies and Contradictions

Examples:

The negation of any tautology is a contradiction, and

the negation of any contradiction is a tautology.



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Exercises

(PQ) (P)(Q)

1.Show that (PQ) (P)(Q).

De Morgans laws

2. Show that p q and pq are logically equivalent

3. Show that (p q) and pq are logically equivalent





logically equivalent

P Q (PQ) (P)(Q) (PQ)(P)(Q)

true true false false true

true false false false true

false true false false true


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Operator Precedence

1

2

3

4

5



Truth

ValueBit

True 1

False 0

Table for the Bit Operators

OR, AND, and XOR.

x y x y x y x y

0 0 0 0 0

0 1 1 0 1

1 0 1 0 1

1 1 1 1 0

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Making Not Gates



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2013-2014 Spring

Discrete Mathematics


482013-2014 Spring Discrete Mathematics


AB

A BA

A B

Y

BInputs Output

A B C Y

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 0

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Examples

492013-2014 Spring Discrete Mathematics




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Predicates and Quantiers -1

cannot adequately express the meaning of all statements in mathematics and in natural language

Every computer connected to the university network is functioning properly.

MATH3 is functioning properly,




CS2 is under attack by an intruder,

There is a computer on the university network that is under attack by an intruder.

predicate logic

Quantiers: all, some, many, none, and few.

all of, for each, given any, for arbitrary, for each and for any.



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The meaning of the universal quantiers

Statement When True? When False?

true

false

true

false

Examples -1

Solution:

true

Solution:

falsefalse



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Predicates -1

true false



Predicates -2

rst part, the variable x

second part - the predicate, is greater than 3

Once a value has been assigned to the variable x, the statement P(x) becomes a proposition and has a truth value



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Examples -1

Solution:

true

false

A(x) Computer x is under attack by an intruder.

Solution:

false true true



Examples -2

Solution:

false

true

A(c,n) Computer c is connected to network n

MATH1 is connected to network CAMPUS2, but not to network CAMPUS1.

Solution:

F

T



1-disc math spring 2014-1

Documents

noits truth value

value of y

type of statement

discrete mathematicsweek

true t2013

false f2013

declarative sentences

eref naci engin office