Logic - Elementary Discrete Math1 Discrete Mathematics Jim Skon Mount Vernon Nazarene College Chapter 1

Logic - Elementary Discrete Math 1

Discrete Mathematics

Jim Skon

Mount Vernon Nazarene College

Chapter 1


Discrete Mathematics Discrete - "Consisting of unconnected

parts"Discrete/Continuous Math

pieces of ice vs. liquid water Letter grade vs. gpa


Discrete MathematicsApplications of discrete mathematics:

Formal Languages (computer languages) Compiler Design Data Structures Computability Automata Theory Algorithm Design Relational Database Theory Complexity Theory (counting)


Discrete MathematicsExample (counting):

The Traveling Salesman Problem Important in

• circuit design• many other CS problems


The Traveling Salesman ProblemGiven:

n cities c1 , c2 , . . . , cn

distance between city i and j, dij

Find the shortest tour.

a

c

b

d

3

54

7

2dab = 3


The Traveling Salesman Problem Assume a very fast PC:1 flop = 1 nanosecond

= 10 -9 sec.= 1,000,000,000 ops/sec= 1 GHz.


The Traveling Salesman ProblemA tour requires n-1 additions. How

many different tours? Choose the first city n ways, the second city n-1 ways, the third city n-2 ways, etc.

a

c

b

d

3

54

7

2


The Traveling Salesman Problem Total number of tours

n (n-1) (n-2) . . . .(2) (1) = n! (Combinations)

Total number of additions (n-1) n! (Rule of Product)


The Traveling Salesman ProblemIf n = 8,

T(n) = 78! = 282,240 flops < 1/3 second.

HOWEVER


The Traveling Salesman ProblemIf n=50,

T(n) = 49 50!= 1.48 10 66

= 1.49 10 57 seconds= 2.48 10 55 minutes= 4.13 10 53 hours= 1.72 10 52 days= 2.46 10 51 weeks= 4.73 10 49 years.


The Traveling Salesman Problem...a long time. You’ll be an old

person before it’s finished.There are some problems for which

we do not know if efficient algorithms exist to solve them!


Propositions An assertion which is either true (or 1) or false (or

0). Today is Wednesday. Some dogs each fish. 4 + 2 = 5 There exists a person who can run 100 mph. The number of hairs on Bill Clinton’s head is 5,543,234,123. Every natural number can be written as the sum of the squares of

four natural numbers. The equation xn + yn = zn where n > 2 and x, y, and z are positive

integers has no solutions. (Fermat’s last Theorem) Every even number greater then 4 can be written as the sum of two

prime numbers.


Propositions Now consider:

• a. Give me my book. (Imperative)

• b. What is my test score? (Interrogative)

• c. Four score and seven years ago (Clause)


Logical ConnectivesConsider

I like rock and I like classical. I’ll either eat at Jody’s or I will go to class. I did not do my homework today. I will either ride will Bill or I will walk

tomorrow, but not both.


Logical ConnectivesProposition - Any sentence with a truth

valueSimple Proposition - No connectivesCompound Proposition - made up of one or

more simple Propositions linked together by connectives.


Logical ConnectivesExamples:

I like pop music and I like classical music. I got an A in English or I got an B in

Philosophy. You did not bring me what I wanted. If you see my mother, then tell her where I am.


Logical Connectives 4 basic connectivesConnective Notation Name of Connective

not negation

and conjunction

or disjunction

if..then.. conditional, implication


Logical ConnectivesSentence form statements may be translated to

symbolic form:• If today is Wednesday and the time is 10:30 am then we

should all be in chapel.

P: today is Wednesday

Q: time is 10:30 am

R: We should all be in chapel

(P Q) R


Logical ConnectivesIt is false that roses are red and violets are

blue. P: roses are red Q: violets are blue

(p q)


Truth Tables:Negation

p p

T F

F T

Negation ‘not’ Symbol:

Example:P: I am going to townP:I am not going to town;It is not the case that I am going to

town;I ain’t goin’.


Truth Tables:

Disjunction

p q p q

T T T

T F T

F T T

F F F

Disjunction inclusive ‘or’ Symbol:

Example: P - ‘I am going to

town’ Q - ‘It is going to

rain’ PQ: ‘I am going to

town or it is going to rain.’


Truth Tables:Conjunction

‘and’ Symbol:

Example:P - ‘I am going to town’Q - ‘It is going to rain’PQ: ‘I am going to town and

it is going to rain.’

Conjunction

p q p qT T T

T F F

F T F

F F F


Truth Tables:Exclusive OR

Symbol:

Example:P - ‘I am going to town’Q - ‘It is going to rain’PQ: ‘Either I am going to town

or it is going to rain.’

Exclusive Or

p q p q

T T F

T F T

F T T

F F F


Truth Tables: Implication

‘If...then...’ Symbol:

Example:P - ‘I am going to town’Q - ‘It is going to rain’P Q: ‘If I am going to town then

it is going to rain.’

Implicationp q p q

T T T

T F F

F T T

F F T


ImplicationEquivalent forms:

If P, then Q P implies Q If P, Q P only if Q P is a sufficient condition for Q Q if P Q whenever P Q is a necessary condition for P

Note: The implication is false only when P is true and Q is false!

Implicationp q p q

T T T

T F F

F T T

F F T


ImplicationThe implication p q is the proposition that is false

when p is true and q is false and true otherwise.p is called the hypothesis (or antecedent or premise)q is called the conclusion (or consequence)Conditional

p q p q

T T T

T F F

F T T

F F T


ImplicationConsider

If it is sunny, class will meet in the grove If today is Saturday, then we will all play leap frog. If x is even then x2 is even If the moon is made of green cheese then I

have more money than Bill Gates If the moon is made of green cheese then I’m

on welfare If 1+1=3 then your grandma wears combat

boots


Implication implication P Q converse Q P inverse P Q contrapositive Q P

QP is the CONVERSE of P QQP is the CONTRAPOSITIVE of P Q


Implication implication P Q

converse Q P

inverse P Q

contrapositive Q P

If it's after 10:00, then I will go to bed.If I go to bed, then it is after 10:00.If it's not after 10:00 then I will not go to bed.If I do not go to bed, then it is not after 10:00.


ImplicationFind the converse and contrapositive of

the following statement:R: ‘Raining tomorrow is a sufficient

condition for my not going to town.’


ImplicationStep 1: Assign propositional

variables to component propositionsP: It will rain tomorrowQ: I will not go to town


ImplicationStep 2: Symbolize the assertion

R: PQ

Step 3: Symbolize the converseQP


ImplicationStep 4: Convert the symbols back into

words‘If I don’t go to town then it will rain

tomorrow’or‘Raining tomorrow is a necessary

condition for my not going to town.’or‘My not going to town is a sufficient

condition for it raining tomorrow.’


Implication implication converse inverse contrapositive

p q p q p q q p p q q p

T T F F T T T T

T F F T F T T F

F T T F T F F T

F F T T T T T T

Therefore: p q q p, implication contrapositive

q p p q, converse inverse


Examples Make a truth table for:

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


Logical ConnectivesPut in symbolic form:

I am not a mathematician. I am not a mathematician and 2 + 2 = 5. If I am not a mathematician then 2 + 2 = 5. If it is raining, then I will not go to the store, and

if it is not raining, then I will go to the store. Either Frank love Mary, or Mary loves Frank, but

not both.


Examples1. MVNC is in Mt. Vernon or Mt. Vernon is in Russia.

2. MVNC is in Ohio and Lake Erie is in Florida.

3. If MVNC is in Ohio then Mt. Vernon is in Ohio.

4. If MVNC is in Ohio then Mt. Vernon is made of cheese.

5. If MVNC is a military school then Mt. Vernon is in Ohio.

6. If MVNC is a military school then Mt. Vernon is made of cheese.


Examples Let p = True, q = False, r = True.

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


Logical Equivalence How about:

If x = 2, then x2 = 4 If x is positive and x2 = 4,

then x = 2 If n2 is odd, then n is odd


Implication p q may be stated as: "if p then q" "p only if q" "p is a sufficient condition for q" "q is a necessary condition for p" "a necessary condition for p is q" "a sufficient condition for q is p" "q if p" "q follows from p" "q is a logical consequence of p" "q whenever p"


Translating English SentencesYou cannot take CS II if you have not

taken CS or have not passed a c proficiency exam

Try 1.1:problem 5


Propositional VariablesA proposition can also be represented as a

variable.• Use lower case for numeric variables.

• Use upper case for propositional variables.

• Example:– Let p be “Today is Wednesday.”

– Let p be “I like to eat in the MVNC cafeteria.”

– Let p be “1 = 3”


BiconditionalLet p and q be propositions.The biconditional p q is the proposition that is

true when p and q have the same truth values, and false otherwise.

It is BOTH the implication AND its converse Ex: “I will eat in the cafeteria if and only if they

have pizza.” biconditionalp q p qT T TT F FF T FF F T


Translating to English sentences p: you like eating candyq: you lose your teeth

p q p q(p q) p q


Logic and Bit Operationsbit - two values: 0 or 1bit can be used to store truth values

0 = false 1 = true

Thus logic operators can apply


Bit operation Truth Tables:Negation

p p

1 0

0 1

Disjunction

p q p q

1 1 1

1 0 1

0 1 1

0 0 0

Conjunction

p q p q1 1 1

1 0 0

0 1 0

0 0 0

Exclusive OR

p q p q

1 1 0

1 0 1

0 1 1

0 0 0


Bit StringsA bit string is a sequence of zero or more

0’s and 1’sExample

0100010010 00111011 1111111111111 00100100000010101 0


Bit Strings Bit operations can be extended to bit stringsexample:

01000101

00111101

00000101 bitwise and

01111101 bitwise or

01111000 bitwise xor


Propositional EquivalencesTautology - true for any values assigned to it's

variablesContradiction - never true, irregardless of

values of variables.Consider

I will either get a new car, or I won’t get a new car. I will come tomorrow and I won’t come tomorrow


Propositional Equivalences Examples

p p p p p p

F T T F

T F T F

Tautology Contradiction


Propositional Equivalences Tautologies are logically true. Contradictions are logically false. Such statements are true or false because of

it's structure rather then because of it's variables truth values.


Propositional EquivalencesConsider:

p (q p)

(p q) (p q)


Logical EquivalenceTwo compound statements are logically

equivalent if they always have the same truth value.

Two compound statements are logically equivalent if they have the same truth tables.

If p and q are logically equivalent, then:

p q


Logical Equivalence Consider:

(p q) (p q)


Logical Equivalence Logical Identities

Useful properties of logic Similar to the properties of Algebra


Logical Equivalence Properties of Identity and Dominance:

p F p , p T p

p T T , p F F

The idempotent properties:

p p p , p p p


Logical EquivalenceThe commutative properties:

p q q p

p q q p

The associative properties:

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


Logical Equivalence The distributive properties:

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

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


Logical Equivalence properties of Complement:

T F , F T

p p T , p p F

(p) p


Logical Equivalence Demorgan's laws:

(p q) p q

(p q) p q


Logical Equivalence Implication

p q p q

Negation of Implication

(p q) p q


Logical Equivalence If and only if

p q (p q) (q p)


Logical Equivalence By Demorgan's law the negation of

I don’t like to eat apples and I don’t like to walk.

becomes

It's not true that I like to eat apples or I don't like to walk.


Logical EquivalenceExamples:

Try 1.2: 7, 9, 11, 13, 15, 17, 19


Logical Equivalence Notice that:

p q (p q) (q p)In order to prove a biconditional statement P Q, we

must only prove the statement

p q, and it's converse

q p.(Or perhaps by proving the inverse and the contrapositive).


Propositions Free Variables

A proposition may contain variable - values not yet assigned.

Variables must be assigned values before the truth value can be assessed.

Examples x + y = 11 Car m the faster then car n. There are x people in this room. I have h hairs on my head.


Propositions as FunctionsA proposition with variables can be written as a

function: x2 > 4 P(x) x + 3y2 + 8 = y + 2x2 - 15 Q(x, y) x is the mother of y M(x, y) Student a is in the b class S(a, b, c)

and lives in apartment c.

How would we write: 32 > 4 Sally is Bill's mother 3 + 3(2)2 + 8 = 5 + 2(3)2 - 15


Existential QuantificationConsider the proposition:

There is a person in this class with a January birthday.

Existentially quantified expression: x : (x is in this class and x has a January

birthday)


Existential QuantificationConsider again:

There is a person in this class with a January birthday.

Existentially quantified expression:

x : ( C(x) JBD(x) )Read "There exists an x such that C(x) and

JBD(x) are true.


Existential Quantification - existential quantifierx : - x is a bound variablex : P - existentially quantified

proposition


Existential QuantificationTo show that an existentially quantified

proposition x : P is true, you must only find a single example of x which makes P true.

To show that an existentially quantified proposition x : P is false, you must show that x possible values of x makes P false.


Universal QuantificationConsider the proposition:

Every student eating in the MVNC cafeteria must either have a meal ticket, or have paid at the door.

Universal quantified expression:

s : if s is eating in the cafeteria s has a meal ticket or s paid at the door.


Universal QuantificationConsider again:

Every student eating in the MVNC cafeteria must either have a meal ticket, or have paid at the door.

Functions: s eating in the cafeteria C(s) s has a meal ticket MT(s) s paid at the door PAID(s)

Existentially quantified expression

s : (C(s) MT(s) PAID(s) ) )


Universal QuantificationEvery student eating in the MVNC

cafeteria must either have a meal ticket, or have paid at the door.

Universally quantified expression

s : (C(s) MT(s) PAID(s) ) )Read "For every student s, if s is eating in the

cafeteria then s has a meal ticket or s has paid at the door.


Universal Quantification - universal quantifierx : - x is a bound variablex : P(x) - universally

quantified proposition


Universal QuantificationTo show that an universally quantified

proposition x : P is true, you must show that P holds for all values of x .

To show that an universally quantified proposition x : P is false, you must only show at least one x for which P is false (a counter example)


Quantification Consider:

For every x, x2 > 4 if and only if x > 2 or x < -2. There exists a number that equals it own square For every number x > 1, there exists a number y where x < y <

2x. There exists a number x such that for every y, y2 > x. Everybody has a mother and a father. Tom and Harry have the same mother. Nobody is their own mother. Peter has no children.


Universal QuantificationThe following ranger over the real numbers:

x:y: x + y = y.x:y: x + y = y.x:y: x + y = y.x:y: x + y = y.


Universal QuantificationThe following range over the integers:

x:y: (x2 = y2 x = -y).

x:y: (x = y x > y).

x:y: (x = y x > y).


Negations of Quantified Propositions Let P be a proposition. Then:

x:P x:P

x:P x:P


Negations of Quantified Propositions

Consider: There is a student in this class with a January

birthday.

x: JB(x) It is not true that there is a a student in this class

with a January birthday.

x: JB(x) x: JB(x)


Negations of Quantified Propositions

Expand domain to include all students:

x: CL(x) JB(x)

x: CL(x) JB(x) )

x: CL(x) JB(x)


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Logic - Elementary Discrete Math1 Discrete Mathematics Jim Skon Mount Vernon Nazarene College Chapter 1