logic and function

7/29/2019 Logic and function

1/62

Module #1 - Logic

1

The Foundations of Logic


2/62

2

Foundations of Logic

Mathematical Logic is a tool for working withelaboratecompoundstatements. It includes:

A formal languagefor expressing them. A concisenotation for writing them.

A methodology for objectively reasoning abouttheir truth or falsity.

It is the foundation for expressingformal proofsin all branches of mathematics.


3/62

3

Foundations of Logic: Overview

Propositional logic:

Basic definitions.

Equivalence rules & derivations. Predicate logic

Predicates.

Quantified predicate expressions. Equivalences & derivations.


4/62

4

Propositional Logic

Propositional Logic is the logic of compoundstatements built from simpler statementsusing so-calledBooleanconnectives/operators.

Some applications in computer science:

Design of digital electronic circuits.

Expressing conditions in programs.

Queries to databases & search engines.

Topic #1 Propositional Logic


5/62

5

Definition of aProposition

Definition: A proposition(denotedp, q, r, ) issimply:

astatement(i.e., a declarative sentence) with somedefinitemeaning,

(not vague or ambiguous)

having atruth value

thats either true(T) or false(F) it isnever both, neither, or somewhere in between!

However, you might notknowthe actual truth value,

and, the value mightdependon the situation or context.



6/62

6

Examples of Propositions

It is raining. (In a given situation.)

Beijing is the capital of China.

1 + 2 = 3But, the following areNOT propositions:

Whos there? (interrogative, question)

Just do it! (imperative, command) 1 + 2 (expression with a non-true/false

value)



7/62

7

Anoperator or connectivecombines one or moreoperandexpressions into a larger expression.

(E.g., + in numeric exprs.) Unaryoperators take 1 operand (e.g., 3);

Binaryoperators take 2 operands (e.g., 3 4).

Propositional or Booleanoperatorsoperateonpropositions (or their truth values) instead ofonnumbers.

Operators / Connectives

Topic #1.0 Propositional Logic: Operators


8/62

8

Some Popular Boolean Operators

Formal Name Nickname Arity Symbol

Negation operator NOT Unary Conjunction operator AND Binary

Disjunction operator OR Binary

Exclusive-OR operator XOR Binary

Implication operator IMPLIES Binary

Biconditional operator IFF Binary



9/62

9

The Negation Operator

The unarynegation operator (NOT)transforms a prop. into its logical negation.

E.g. Ifp= I have brown hair.then p= I donot have brown hair.

Thetruth tablefor NOT: p p

T F

F TT : True; F : False: means is defined as

Operandcolumn

Resultcolumn



10/62

10

The Conjunction Operator

The binaryconjunction operator (AND)combines two propositions to form their

logical conjunction.E.g. Ifp=I will have salad for lunch. and

q=I will have steak for dinner., then

pq=I will have salad for lunchandI will have steak for dinner.

Remember: points up like an A, and it means ND

ND



11/62

11

Note that aconjunctionp1 p2 pnofnpropositionswill have 2n rows

in its truth table.

Conjunction Truth Table

p q pq

F F FF T F

T F F

T T T

Operand columns



12/62

12

The Disjunction Operator

The binarydisjunction operator (OR)combines two propositions to form their

logical disjunction.p=My car has a bad engine.

q=My car has a bad carburetor.

pq=Either my car has a bad engine, ormy car has a bad carburetor.


Meaning is like and/or in English.


13/62

13

Note thatpqmeansthatp is true, or q is

true, or both are true! So, this operation is

also called inclusive or,

because it includesthepossibility that bothpandqare true.

Disjunction Truth Table

p q pq

F F FF T T

T F T

T T T

differencefrom AND



14/62

14

Nested Propositional Expressions

Use parentheses togroup sub-expressions:I just saw my oldfriend, and either hes

grownor Iveshrunk. =f

(g

s) (f g) s would mean something

different

f g s would be ambiguous

By convention, takesprecedenceoverboth and .

s f means (s) f , not (s f)



15/62

15

A Simple Exercise

Letp= It rained last night,q= The sprinklers came on last night,

r = The lawn was wet this morning.Translate each of the following into English:

p =

r p =r p q =

It didnt rain last night.

The lawn was wet this morning, andit didnt rain last night.

Either the lawn wasnt wet thismorning, or it rained last night, orthe sprinklers came on last night.



16/62

16

TheExclusive Or Operator

The binaryexclusive-or operator (XOR)combines two propositions to form their

logical exclusive or.p= I will earn an A in this course,

q=I will drop this course,

p q= I will either earn an A in this course,or I will drop it (but not both!)



17/62

17

Note thatpqmeansthatp is true, or q is

true, butnot both! This operation is

calledexclusive or,

because itexcludesthepossibility that bothpandqare true.

Exclusive-Or Truth Table

p q pq

F F FF T T

T F T

T T F Notedifferencefrom OR.



18/62

18

Note that Englishor can be ambiguousregarding the both case!

Pat is a singer orPat is a writer. -

Pat is a man or

Pat is a woman. -Need context to disambiguate the meaning!

For this course, assume or means inclusive.

Natural Language is Ambiguous

p q p"or" q

F F F

F T T

T F T

T T ?



19/62

19

TheImplicationOperator

Theimplicationp qstates thatp impliesq.

i.e., Ifp is true, thenq is true; but ifp is nottrue, thenqcould be either true or false.

E.g., letp= You study hard.q= You will get a good grade.

p q =If you study hard, then you will geta good grade. (else, it could go either way)

antecedent consequent


20/62

20

Implication Truth Table

p q is falseonlywhenp is true butq isnot true.

p q doesnot saythatpcausesq!

p q doesnot require

thatpor qare ever true! E.g. (1=0) pigs can fly is TRUE!

p q pq

F F T

F T T

T F F

T T T

TheonlyFalse

case!



21/62

21

Examples of Implications

If this lecture ends, then the sun will risetomorrow.Trueor False?

If Tuesday is a day of the week, then I am apenguin.Trueor False?

If 1+1=6, then Obama is president of USA.

Trueor False? If the moon is made of green cheese, then I

am richer than Bill Gates.Trueor False?



22/62

22

English Phrases Meaningp q

p impliesq

ifp, thenq

ifp, q whenp, q

whenever p, q

q

ifp

qwhenp

qwhenever p

ponly ifq

p is sufficient for q

q is necessary for p q follows fromp

q is implied by p

We will see some equivalentlogic expressions later.

i i i l i


23/62

23

Converse, Inverse, Contrapositive

Some terminology, for an implicationp q:

Itsconverseis: q p.

Its inverse is: p q. Itscontrapositive: q p.

One of these three has thesame meaning

(same truth table) asp q. Can youfigure out which?


T i #10 P i i l L i O


24/62

24

How do we know for sure?

Proving the equivalence ofp qand itscontrapositive using truth tables:

p q q p pq qp

F F T T T T

F T F T T T

T F T F F FT T F F T T


T i #10 P iti l L i O t


25/62

25

Thebiconditional operator

Thebiconditional p qstates thatp is trueif and

only if (IFF) q is true.p= You can take the flight.

q=You buy a ticket.

p q =You can take the flight if and only if youbuy a ticket.




26/62

26

Biconditional Truth Table

p qmeans thatpandqhave thesametruth value.

Note this truth table is theexactoppositeofs!

Thus, p qmeans (p q)

p qdoesnot implythatpandqare true, or cause each other.

p q pq

F F T

F T F

T F F

T T T




27/62

27

Boolean Operations Summary

We have seen 1 unary operator and 5binary operators. Their truth tables are

below.p q p pq pq pq pq pq

F F T F F F T T

F T T F T T T FT F F F T T F F

T T F T T F T T


T i #2 Bit


28/62

28

Bits and Bit Operations

A bit is a binary (base 2) digit: 0 or 1.

Bits may be used to represent truth values.

By convention:0 represents false; 1 represents true.

Boolean algebra is like ordinary algebra

except thatvariables stand for bits,+ means or, andmultiplication means and.

Topic #2 Bits

Topic#2 Bits


29/62

29

Bit Strings

A Bit stringof length n is anordered seriesor sequence of n 0 bits.

By convention, bit strings are written left toright: e.g. the first bit of 1001101010 is 1.

When a bit string represents a base-2 number,by convention the first bit is themost

significantbit. Ex. 11012=8+4+1=13.

Topic #2 Bits

Topic#2 Bits


30/62

30

Bitwise Operations

Boolean operations can be extended tooperate on bit strings as well as on single bits.

E.g.:01 1011 011011 0001 1101

11 1011 1111 Bit-wise OR01 0001 0100 Bit-wise AND10 1010 1011 Bit-wiseXOR

Topic #2 Bits


31/62

31

Summary on Propositional Logic

You have learned about:

Propositions: Whatthey are.

Propositional logicoperators Symbolic notations.

English equivalents.

Logical meaning.

Truth tables.

Atomic vs. compoundpropositions..

Bits and bit-strings.

Upcoming topics:

Propositional

equivalences. How to prove them.

Topic#11 Propositional Logic: Equivalences


32/62

32

Propositional Equivalence

Twosyntactically(i.e., textually) differentcompound propositions may be

semantically identical (i.e., have the samemeaning). We call themequivalent.

Learn:

Variousequivalence rulesor laws. How toproveequivalences usingsymbolic

derivations.

Topic #1.1 Propositional Logic: Equivalences



33/62

33

Tautologies and Contradictions

A tautology is a compound proposition that istrueno matter what the truth values of itsatomic propositions are!

Ex. p p [What is its truth table?]

A contradiction is a compound propositionthat is falseno matter what! Ex. p p

[Truth table?]Other compound props. arecontingencies.




34/62

34

Logical Equivalence

Compound propositionp is logicallyequivalent to compound propositionq,

writtenp

q, IFF the compoundpropositionpq is a tautology.

Compound propositionspandqare logically

equivalent to each other IFF pandqcontain the same truth values as each otherin all rows of their truth tables.




35/62

35

Ex. Prove thatpq (p q).

p q ppqq pp qq ppqq ((ppqq))

F F

F T

T F

T T

Proving Equivalence

via Truth Tables

FT

TT

T

T

T

TT

T

FF

F

F

FF

FF

TT




36/62

36

Equivalence Laws

These are similar to the arithmeticidentities you may have learned in algebra,

but for propositional equivalences instead. They provide a pattern or template that can

be used to match all or part of a much more

complicated proposition and to find anequivalence for it.




37/62

37

Equivalence Laws - Examples

Identity: pT p pF p

Domination: pT T pF F

Idempotent: pp p pp p Double negation: p p

Commutative: pq qp pq qp

Associative: (pq)r p(qr)(pq)r p(qr)




38/62

38

More Equivalence Laws

Distributive: p(qr) (pq)(pr)p(qr) (pq)(pr)

De Morgans:(pq) p q(pq) p q

Trivial tautology/contradiction:

p p T p p F


Augustus

De Morgan(1806-1871)



39/62

39

Defining Operators via Equivalences

Using equivalences, we candefineoperatorsin terms of other operators.

Exclusive or: pq (pq)(pq)pq (pq)(qp)

Implies: pq p q

Biconditional: pq (pq) (qp)pq (pq)




40/62

40

An Example Problem

Check using a symbolic derivation whether (p ( p q)) p q.



41/62

41

Predicate Logic

Predicate logic is an extension ofpropositional logic that permits concise

reasoning about wholeclassesof entities. Propositional logic (recall) treats simple

propositions (sentences) as atomic entities.

In contrast, predicatelogic distinguishesthesubjectof a sentence from itspredicate.

Remember these English grammar terms?

Topic#3 PredicateLogic


42/62

42

Subjects and Predicates

InthesentenceThedogissleeping:

The phrase the dog denotes thesubject -

theobject

orentity

that the sentence is about. The phrase is sleeping denotes the

predicate- a property that is trueofthe subject.

In predicate logic, apredicateis modeled asafunctionP() from entities to propositions.

P(x) = x is sleeping (wherex is any object).

Topic #3 Predicate Logic



43/62

43

More About Predicates

Convention: Lowercase variablesx, y, z... denoteobjects/entities; uppercase variablesP, Q, Rdenote propositional functions (predicates).

Keep in mind that theresult ofapplyingapredicateP to an objectx is theproposition P(x).But the predicateP itself(e.g. P=is sleeping) is

not a proposition (not a complete sentence). E.g. ifP(x) = x is a prime number,

P(3) is theproposition3 is a prime number.




44/62

44

Propositional Functions

Predicate logicgeneralizesthe grammaticalnotion of a predicate to also include

propositional functions ofanynumber ofarguments, each of which may takeanygrammatical role that a noun can take.

E.g. letP(x,y,z) = xgaveythe gradez, thenifx=Mike, y=Mary, z=A, thenP(x,y,z) =Mike gave Mary the grade A.




45/62

45

Universes of Discourse (U.D)

The power of distinguishing objects frompredicates is that it lets you state things

aboutmanyobjects at once. E.g., letP(x)=x+1>x. We can then say,

Foranynumberx, P(x) is true instead of

(0+1>0) (1+1>1) (2+1>2) ... The collection of values that a variablexcan

take is calledxsuniverse of discourse.




46/62

46

Quantifier Expressions

Quantifiersprovide a notation that allowsus toquantify(count) how manyobjects inthe univ. of disc. satisfy a given predicate.

is the FORLL or universalquantifier.xP(x) means for all x in the u.d., P holds.

is theXISTS or existential quantifier.x P(x) means thereexistsanx in the u.d.(that is, 1 or more) such thatP(x) is true.

p g



47/62

47

The Universal Quantifier

Example:Let the u.d. of x be parking spaces at UBC.LetP(x) be thepredicatex is full.

Then theuniversal quantification of P(x),xP(x), is theproposition: All parking spaces at UBC are full.

i.e., Every parking space at UBC is full. i.e., For each parking space at UBC, that

space is full.

p g



48/62

48

The Existential Quantifier

Example:Let the u.d. of x be parking spaces at UBC.

LetP(x) be thepredicatex is full.Then theexistential quantification of P(x),xP(x), is theproposition:

Some parking space at UBC is full.

There is a parking space at UBC that is full.

At least one parking space at UBC is full.

p g



49/62

49

Free and Bound Variables

An expression likeP(x) is said to have afree variablex(meaning, x is undefined).

A quantifier (either or ) operateson anexpression having one or more freevariables, andbindsone or more of thosevariables, to produce an expression havingone or moreboundvariables.

p g



50/62

50

Example of Binding

P(x,y) has 2 free variables, xandy.

xP(x,y) has 1 free variable, and one boundvariable. [Which is which?]

P(x), wherex=3 is another way to bindx.

An expression with zero free variables is anactual proposition.

An expression with one or morefree variables isstill only a predicate: e.g. letQ(y) =xP(x,y)

p g


51/62

51

Negations on Quantifiers

We will often want to consider the negationof a quantified expression.

Example: Consider the statement Every student in the

class has taken a course in calculus.

This statement is a universal quantification,namely, x P(x) where P(x) is the statementx has taken a course in calculus


52/62

52

Negate a Universal Quantification

The negation of the above statement is

It is not the case that every student has taken a

course in calculus, namely, x P(x). Or, put it another way,

There is at least a student in the class who has nottaken a course in calculus, namely, xP(x).

This example illustrates the following equivalence:

x P(x) xP(x)


53/62

53

Negate an Existential Quantification

Example: Consider the statement

There is a student in the class who has taken acourse in calculus, namely x Q(x), where Q(x) is

the statement x has a course in calculus. The negation of this statement is the proposition

It is not the case that there is a student in theclass who has taken a course in calculus,namely, x Q(x).

This is equivalent toEvery student in this class has not taken a coursein calculus, namely, x Q(x).

So x Q(x) x Q(x).


54/62

54

Negation Equivalence

De Morgans Laws in the case of negations ofquantifiers (assuming that all the elements of u.d.can be listed)

x P(x) (P(x1) P(x2) P(xn)) P(x1) P(x2) P(xn)

x P(x)

x P(x) (P(x1) P(x2) P(xn)) P(x1) P(x2) P(xn)

x P(x)


55/62

55

Examples on Negations

What is the negations of the followingstatement? There is an honest politician

Solution: There is an honest politician is represented

by x H(x) where H(x) is the statement x is

an honest politician The negation is

There is not a single honest politicianwhich is represented by

x H(x), or x H(x).


56/62

56

Examples on Negations (cont.)

What are the negations of the followingstatements? All Canadians play hockey

Solution: All Canadians play hockey is represented

by x H(x), where H(x) is the statement

x plays hockey The negation is

Some Canadian does not play hockey,which is represented by

x H(x), or x H(x).



57/62

57

Nesting of Quantifiers

Example: Let the u.d. ofx& ybe people.

LetL(x,y)=x likesy (a predicate w. 2 f.v.s)

Theny L(x,y) = There is someone whomxlikes. (A predicate w. 1 free variable, x)

Thenx(y L(x,y)) =

Everyone has someone whom they like.(A __________ with ___ free variables.)



58/62

58

Quantifier Exercise

IfR(x,y)=xrelies upony,express the following in unambiguous English:

x(y R(x,y))=y(xR(x,y))=

x(y R(x,y))=

y(x R(x,y))=x(yR(x,y))=

Everyone hassomeoneto rely on.

Theres a poor overburdened soul whomeveryonerelies upon (including himself)!

Theres some needy person who relies

uponeverybody(including himself).Everyone hassomeonewho relies upon them.

Everyonerelies uponeverybody,(including themselves)!



59/62

59

Some Conventions

Sometimes the universe of discourse isrestricted within the quantification, e.g.,

x>0P(x) is shorthand forFor all xthat are greater than zero, P(x).

=x(x>0 P(x))

x>0P(x) is shorthand for

There is anxgreater than zero such thatP(x).

=x(x>0 P(x))



60/62

60

Defining New Quantifiers

As per their name, quantifiers can be used toexpress that a predicate is true of any givenquantity

(number) of objects.Define!xP(x) to mean P(x) is true ofexactly onex in the universe of discourse.

!xP(x) x(P(x) y(P(y) yx))There is anxsuch thatP(x), where there isnoysuch that P(y) andy is other thanx.



61/62

61

A Number Theory Example

Let u.d. = the positiveinteger numbers0, 1, 2,

A number x iseven, E(x), if and only if

it is equal to 2 times some other number.x(E(x) (y x=2y))

A number isprime, P(x), iff

its greater than 1 and it isnt the product of any

two non-unity numbers.x(P(x) (x>1 yz x=yz y1 z1))



62/62

Review: Predicate Logic

You should have learned:

Predicate logic notation & conventions

Conversions: predicate logic

clear English Meaning of quantifiers, equivalences

Upcoming topics:

Introduction to proof-writing. Then: Set theory

a language for talking about collections of objects.

logic and function

Documents