discreet mathematics

Logic!

Jan-May 2013 Introduction to Mathematical Logic: MATH-105 1

• Crucial for mathematical reasoning• Important for program design• Used for designing electronic circuitry

• (Propositional )Logic is a system based on propositions.

• A proposition or statement is a (declarative) sentence that is either true or false (not both).

• We say that the truth value of a proposition is either true (T) or false (F).

• Corresponds to 1 and 0 in digital circuits


“Elephants are bigger than mice.”


Is this a sentence?Is this a sentence? yesyes

Is this a proposition?Is this a proposition? yesyes

What is the truth What is the truth value value

of the proposition?of the proposition?truetrue

“520 < 111”





of the proposition?of the proposition?falsfals

ee

“y > 5”



Is this a proposition?Is this a proposition? nono

Its truth value depends on the value of Its truth value depends on the value of y, but this value is not specified.y, but this value is not specified.

We call this type of sentence a We call this type of sentence a propositional functionpropositional function or or open open sentencesentence..

“Today is January 27 and 99 < 5.”





of the proposition?of the proposition?falsfals

ee

“Please do not fall asleep.”



Is this a proposition?Is this a proposition? nono

It’s a request.It’s a request.

“If the moon is made of cheese,then I will be rich.”





of the proposition?of the proposition?probably trueprobably true

“x < y if and only if y > x.”





of the proposition?of the proposition?truetrue

… … because its truth value because its truth value does not depend on does not depend on specific values of x and specific values of x and y.y.

As we have seen in the previous examples, one or more propositions can be combined to form a single compound proposition.

We formalize this by denoting propositions with letters such as p, q, r, s, and introducing several logical operators or logical connectives.


We will examine the following logical operators:

• Negation (NOT, )• Conjunction (AND, )• Disjunction (OR, )• Exclusive-or (XOR, )• Implication (if – then, )• Biconditional (if and only if, )

Truth tables can be used to show how these operators can combine propositions to compound propositions.


Unary Operator, Symbol:


PP PP

true (T)true (T) false (F)false (F)

false (F)false (F) true (T)true (T)

Binary Operator, Symbol:


PP QQ P QP Q

TT TT TT

TT FF FF

FF TT FF

FF FF FF



PP QQ P P QQ

TT TT TT

TT FF TT

FF TT TT

FF FF FF



PP QQ PPQQ

TT TT FF

TT FF TT

FF TT TT

FF FF FF



PP QQ PPQQ

TT TT TT

TT FF FF

FF TT TT

FF FF TT



PP QQ PPQQ

TT TT TT

TT FF FF

FF TT FF

FF FF TT

Statements and operators can be combined in any way to form new statements.


PP QQ PP QQ (P)(Q)(P)(Q)

TT TT FF FF FF

TT FF FF TT TT

FF TT TT FF TT

FF FF TT TT TT

Statements and operators can be combined in any way to form new statements.


PP QQ PQPQ (PQ)(PQ) (P)(Q)(P)(Q)

TT TT TT FF FF

TT FF FF TT TT

FF TT FF TT TT

FF FF FF TT TT

• To take discrete mathematics, you must have taken calculus or a course in computer science.

• When you buy a new car from Acme Motor Company, you get $2000 back in cash or a 2% car loan.

• School is closed if more than 2 feet of snow falls or if the wind chill is below -100.


› P: take discrete mathematics

› Q: take calculus

› R: take a course in computer science

• P Q R

• Problem with proposition R


•To take discrete mathematics, you must To take discrete mathematics, you must have taken calculus or a course in have taken calculus or a course in computer science.computer science.

› P: buy a car from Acme Motor Company

› Q: get $2000 cash back

› R: get a 2% car loan

• P Q R

• Why use XOR here? – example of ambiguity of natural languages


•When you buy a new car from Acme Motor When you buy a new car from Acme Motor Company, you get $2000 back in cash or a Company, you get $2000 back in cash or a 2% car loan.2% car loan.

› P: School is closed

› Q: 2 feet of snow falls

› R: wind chill is below -100

• Q R P

• Precedence among operators: , , , ,


•School is closed if more than 2 feet of School is closed if more than 2 feet of snow falls or if the wind chill is below -100.snow falls or if the wind chill is below -100.

The statements (PQ) and (P) (Q) are logically equivalent, since they have the same truth table, or put it in another way,

(PQ) (P) (Q) is always true.


PP QQ (PQ)(PQ) (P)(Q)(P)(Q) (PQ)(PQ)(P)(Q)(P)(Q)

TT TT FF FF TT

TT FF TT TT TT

FF TT TT TT TT

FF FF TT TT TT

A tautology is a statement that is always true. Examples:

› R(R)› (PQ) (P)( Q)

A contradiction is a statement that is always false. Examples:

› R(R)› ((P Q) (P) (Q))

The negation of any tautology is a contradiction, and the negation of any contradiction is a tautology.

If A has the truth value, T for at least one combination of truth values of P1, P2 , P3 ,…, Pn then A is said to be satisfiable.


Definition: two propositional statements S1 and S2 are said to be (logically) equivalent, denoted S1 S2 if

› They have the same truth table, or› S1 S2 is a tautology

Equivalence can be established by› Constructing truth tables› Using equivalence laws


Equivalence laws› Identity laws, P T P,

› Domination laws, P F F,

› Idempotent laws, P P P,

› Double negation law, ( P) P

› Commutative laws, P Q Q P,

› Associative laws, P (Q R) (P Q) R,

› Distributive laws, P (Q R) (P Q) (P R), › De Morgan’s laws, (PQ) ( P) ( Q)› Law with implication P Q P Q


• Show that P Q P Q: by truth table

• Show that (P Q) (P R) P (Q R): by equivalence laws:

› Law with implication on both sides› Distribution law on LHS


•Proposition› Statement, Truth value, › Proposition, Propositional symbol, Open proposition

•Operators › Define by truth tables› Composite propositions› Tautology and contradiction

•Equivalence of propositional statements› Definition › Proving equivalence (by truth table or equivalence

laws)


Propositional function (open sentence):statement involving one or more variables, e.g.: x-3 > 5.Let us call this propositional function P(x), where P is the predicate and x is the variable.


What is the truth value of P(2) ?What is the truth value of P(2) ? falsefalse

What is the truth value of P(8) ?What is the truth value of P(8) ?

What is the truth value of P(9) ?What is the truth value of P(9) ?

falsefalse

truetrueWhen a variable is given a value, it is said When a variable is given a value, it is said to be instantiatedto be instantiated

Truth value depends on value of Truth value depends on value of variablevariable

Let us consider the propositional function Q(x, y, z) defined as:

x + y = z.Here, Q is the predicate and x, y, and z are the variables.


What is the truth value of Q(2, 3, 5) What is the truth value of Q(2, 3, 5) ??

truetrue

What is the truth value of Q(0, 1, What is the truth value of Q(0, 1, 2) ?2) ?What is the truth value of Q(9, -9, 0) ?What is the truth value of Q(9, -9, 0) ?

falsefalsetruetrue

A propositional function (predicate) becomes a A propositional function (predicate) becomes a proposition when proposition when allall its variables are its variables are instantiatedinstantiated..

Other examples of propositional functions

Person(x), which is true if x is a person


Person(Socrates) = T Person(Socrates) = T

CSCourse(x),CSCourse(x), which is true if x is a which is true if x is a computer science coursecomputer science course

CS Course(C-language) = TCS Course(C-language) = T

Person(dog) = FPerson(dog) = F

CS Course(MATH 102) = FCS Course(MATH 102) = F

Let P(x) be a predicate (propositional function).

Universally quantified sentence:For all x in the universe of discourse P(x) is true.

Using the universal quantifier :x P(x) “for all x P(x)” or “for every x P(x)”

(Note: x P(x) is either true or false, so it is a proposition, not a propositional function.)


Example: Let the universe of discourse be all peopleS(x): x is a Mathematics student.G(x): x is a genius.

What does x (S(x) G(x)) mean ?

“If x is a mathematics student, then x is a genius.” or“All Mathematics students are geniuses.”

If the universe of discourse is all mathematics students, then the same statement can be written as

x G(x)


Existentially quantified sentence:There exists an x in the universe of discourse for which P(x) is true.

Using the existential quantifier :x P(x) “There is an x such that P(x).”

“There is at least one x such that P(x).”

(Note: x P(x) is either true or false, so it is a proposition, but no propositional function.)


Example: P(x): x is a Mathematics professor.G(x): x is a genius.

What does x (P(x) G(x)) mean ?

“There is an x such that x is a Mathematics professor and x is a genius.” or“At least one Mathematics professor is a genius.”


Another example:Let the universe of discourse be the real numbers.

What does xy (x + y = 320) mean ?

“For every x there exists a y so that x + y = 320.”


Is it true?Is it true?

Is it true for the natural Is it true for the natural numbers?numbers?

yesyes

nono

A counterexample to x P(x) is an object c so that P(c) is false.

Statements such as x (P(x) Q(x)) can be disproved by simply providing a counterexample.


Statement: “All birds can fly.”Statement: “All birds can fly.”Disproved by counterexample: Penguin.Disproved by counterexample: Penguin.

(x P(x)) is logically equivalent to x (P(x)).

(x P(x)) is logically equivalent to x (P(x)).

This is de Morgan’s law for quantifiers


Examples

Not all roses are red x (Rose(x) Red(x)) x (Rose(x) Red(x))


Nobody is perfectNobody is perfect x (Person(x)x (Person(x) Perfect(x) Perfect(x)))

x (Person(x)x (Person(x) Perfect(x)Perfect(x)))

A predicate can have more than one variables.› S(x, y, z): z is the sum of x and y› F(x, y): x and y are friends

We can quantify individual variables in different ways› x, y, z (S(x, y, z) (x <= z y <= z))› x y z (F(x, y) F(x, z) (y != z) F(y, z)


Exercise: translate the following English sentence into logical expression“There is a rational number in between

every pair of distinct rational numbers”

Use predicate Q(x), which is true when x is a rational number

x,y (Q(x) Q (y) (x < y) u (Q(u) (x < u) (u < y)))


• Propositional functions (predicates)• Universal and existential quantifiers,

and the duality of the two• When predicates become propositions

› All of its variables are instantiated› All of its variables are quantified

• Nested quantifiers› Quantifiers with negation

• Logical expressions formed by predicates, operators, and quantifiers


Mathematical Reasoning


We need mathematical reasoning to determine whether a mathematical argument is correct or incorrect and construct mathematical arguments.


Argument: An argument is an assertion; that a group of propositions called premises, yields an other proposition, called the conclusion.

Let P1, P2 , P3 ,…, Pn is the group of propositions that yields the conclusion Q. Then, it is denoted as P1, P2 , P3 ,…, Pn |- Q

Valid Argument: An argument is called valid

argument if the conclusion is true whenever all the premises are true orThe argument is valid iff the ANDing of the group of propositions implies conclusion is a tautologyie: P(P1, P2 , P3 ,…, Pn ) Q is a tautology


Fallacy Argument: An argument is called fallacy or an invalid argument if it is not a valid argument.


Translate the following into symbolic form and test the validity of the argument:

1.“ If I will select in IAS examination, then I will not be able to go to London. Since, I am going to London, I will not select in IAS examination”.

2. “If 6 is even then 2 does not divide 7. Either 5 is not prime or 2 divides 7. But 5 is prime, therefore, 6 is not even”.


Ans-1: p: I will select in IAS examinationq: I will go to London Premises: 1. p ~q 2. q

Conclusion: ~p

Verify: [(p ~q) (q)] ~p is a tautology


Ans-2:p: 6 is evenq: 2 divides 7r: 5 is prime

Premises: 1. p ~q 2. ~r q 3. rConclusion: ~p

Verify: [ (1) (2) (3) ] ~p is a tautology


The problem of finding whether a given statement is tautology or contradiction or satisfiable in a finite number of steps, is called as decision problem, the construction of truth tables may not be often a practical solution. We therefore consider alternate procedure known as reduction to normal forms. The two such forms are:

Disjunctive Normal Form (DNF) Conjunctive Normal Form (CNF)


A compound statement is said to be in dnf if it is disjunction of conjunction of the variables or their negations.

Ex:(p q r) (~p q r) (~p ~q r)

A compound statement is said to be in cnf if it is conjunction of disjunction of the variables or their negations.

Ex:(p q ~r) (p ~q r) (~p ~q r)


The truth table for the given statement is:


P Q P Q

Q P

(P Q) (Q P)

T T T T T

T F F T F

F T T F F

F F F T T

DNF:

DNF: (P Q) (~P ~Q)


CNF:

CNF: ~(P ~Q) ~(~P Q) ≡ (~P Q) (P ~Q)


• Mathematical Reasoning• Terminology (argument, valid argument

and fallacy argument)• Examples on valid arguments• Normal forms (DNF and CNF)• Truth table method to express the

statements as dnf and cnf forms


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