discreet mathematics
description
Transcript of 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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 2
“Elephants are bigger than mice.”
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 3
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”
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 4
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?falsfals
ee
“y > 5”
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 5
Is this a sentence?Is this a sentence? yesyes
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.”
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 6
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?falsfals
ee
“Please do not fall asleep.”
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 7
Is this a sentence?Is this a sentence? yesyes
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.”
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 8
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?probably trueprobably true
“x < y if and only if y > x.”
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 9
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
… … 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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 10
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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 11
Unary Operator, Symbol:
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 12
PP PP
true (T)true (T) false (F)false (F)
false (F)false (F) true (T)true (T)
Binary Operator, Symbol:
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 13
PP QQ P QP Q
TT TT TT
TT FF FF
FF TT FF
FF FF FF
Binary Operator, Symbol:
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 14
PP QQ P P QQ
TT TT TT
TT FF TT
FF TT TT
FF FF FF
Binary Operator, Symbol:
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 15
PP QQ PPQQ
TT TT FF
TT FF TT
FF TT TT
FF FF FF
Binary Operator, Symbol:
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 16
PP QQ PPQQ
TT TT TT
TT FF FF
FF TT TT
FF FF TT
Binary Operator, Symbol:
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 17
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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 18
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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 19
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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 20
› P: take discrete mathematics
› Q: take calculus
› R: take a course in computer science
• P Q R
• Problem with proposition R
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 21
•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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 22
•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: , , , ,
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 23
•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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 24
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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 25
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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 26
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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 27
• 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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 28
•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)
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 29
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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 30
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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 31
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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 32
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.)
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 33
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)
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 34
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.)
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 35
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.”
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 36
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.”
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 37
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.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 38
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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 39
Examples
Not all roses are red x (Rose(x) Red(x)) x (Rose(x) Red(x))
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 40
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)
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 41
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)))
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 42
• 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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 43
Mathematical Reasoning
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 44
We need mathematical reasoning to determine whether a mathematical argument is correct or incorrect and construct mathematical arguments.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 45
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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 46
Fallacy Argument: An argument is called fallacy or an invalid argument if it is not a valid argument.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 47
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”.
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 48
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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 49
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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 50
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)
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 51
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)
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 52
The truth table for the given statement is:
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 53
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)
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 54
CNF:
CNF: ~(P ~Q) ~(~P Q) ≡ (~P Q) (P ~Q)
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 55
• 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
Jan-May 2013 Introduction to Mathematical Logic: MATH-105 56