csc331 week 1 topic a

8/13/2019 CSC331 Week 1 Topic A

1/54


2/54

Topic ASets and Logic

Section 1.1Sets

Section 1.2Proposition

Section 1.3Conditional Propositions andLogical Equivalence

Section 1.4Arguments and Rules of

Inference

Section 1.5Quantifiers

Section 1.6Nested Quantifiers


3/54

Section 1.1 Sets

Set = a collection of distinct unordered

objects

Members of a set are called elements How to determine a set

Listing:

Example: A = {1,3,5,7}

Description

Example: B = {x | x = 2k + 1, 0 < k < 3}


4/54

Finite and infinite sets

Finitesets

Examples:

A = {1, 2, 3, 4}

B = {x | x is an integer, 1 < x < 4}

Infinitesets

Examples:

Z = {integers} = {, -3, -2, -1, 0, 1, 2, 3,} S={x| x is a real number and 1 < x < 4} = [0, 4]


5/54

Some important sets

The emptyset has no elements.

Also called null setor void set.

Universalset: the set of all elements about

which we make assertions.

Examples:

U = {all natural numbers}

U = {all real numbers} U = {x| x is a natural number and 1< x


6/54

Some important sets

Two sets X and Y are equal and we write X = Y

if X and Y have the same elements.

X = Y if the following two conditions holds:

For every x, if x X, then x Y,

and

For every x, if x Y, then x X.

This is a great way to prove two sets are equal.


7/54

Cardinality

Cardinality of a set A (in symbols |A|) is the

number of elements in A

Examples:

If A = {1, 2, 3} then |A| = 3

If B = {x | x is a natural number and 1< x< 9}

then |B| = 9

Infinite cardinality Countable (e.g., natural numbers, integers)

Uncountable (e.g., real numbers)


8/54

Subsets

X is a subsetof Y if every element of

X is also contained in Y

(in symbols X

Y) Equality: X = Y if X Y and Y X

X is aproper subsetof Y if X Y but

Y X Observation:is a subset of every set


9/54


10/54

Set operations:

Union and Intersection

Given two sets X and Y

The unionof X and Y is defined as the set

X Y = { x | x X or x Y}

The intersectionof X and Y is defined as the set

X Y = { x | x X and x Y}

Two sets X and Y are disjointif X Y =


11/54

Complement and Difference

The differenceof two sets

XY = { x | x X and x Y}

The difference is also called the relative complement

of Y in X

Symmetric difference

XY = (XY) (YX)

The complement of a set A contained in auniversal set U is the set Ac= UA

In symbols Ac = U - A


12/54

Venn diagrams

A Venn diagram provides a graphic view ofsets

Set union, intersection, difference,

symmetric difference and complements canbe identified


13/54

Properties of set operations (1)

Theorem 2.1.10: Let U be a universal set, and

A, B and C subsets of U. The followingproperties hold:

a) Associativity: (A B) C = A (B C)

(A B) C = A (B C)

b) Commutativity: A B = B A

A B = B A


14/54


c) Distributive laws:

A(BC) = (AB)(AC)A(BC) = (AB)(AC)

d) Identity laws:

AU=A A= A

e) Complement laws:

AAc= U AAc=

Correction!!!!!It was incorrectly reversed in

previous version


15/54


f) Idempotent laws:

AA = A AA = Ag) Bound laws:

AU = U A=

h) Absorption laws:A(AB) = A A(AB) = A


16/54


i) Involution law: (Ac)c= A

j) 0/1 laws: c= U Uc=

k) De Morgans laws for sets:

(AB)c

= Ac

Bc

(AB)c= AcBc


17/54

Union and intersection of

a family Sof sets

The unionof an arbitrary family Sof sets is

defined to be those elements x belonging to

at least one set X in S.

S = {x | x X for some X S}

The intersectionof an arbitrary family Sof

sets is defined to be those elements x

belonging to every set X in SS = {x | x X for all X S}


18/54

Union and intersection of

a family Sof sets (2)


19/54

Partition

Thepartitionof a set X divides X into non-

overlapping subsets.

More formally, a collection S of nonempty

subsets of X is said to be apartition of set X ifevery element in X belongs to exactly one

member of S

If S is a partition of X S is pair-wise disjoint and

S= X


20/54

Logic

Logic = the study of correct reasoning

Use of logic In mathematics:

to prove theorems

In computer science:

to prove that programs do what they are

supposed to do


21/54

Section 1.2 Propositions

Apropositionis a statement or sentence

that can be determined to be either true orfalse.

Examples:

John is a programmer" is a proposition I wish I were wise is not a proposition


22/54

Truth table of conjunction

The truth values of compound propositionscan be described by truth tables.

Truth table of conjunction

p ^ q is true only when both p and q are true.

p q p ^ q

T T T

T F F

F T F

F F F


23/54

Example

Let p = Tigers are wild animals

Let q = Chicago is the capital of Illinois p ^q = "Tigers are wild animals and

Chicago is the capital of Illinois"

p ^q is false. Why?


24/54

Truth table of disjunction

The truth table of (inclusive) disjunctionis

p q is false only when both p and q are false Example: p = "John is a programmer", q = "Mary is a lawyer"

p v q = "John is a programmer or Mary is a lawyer"

p q p v q

T T T

T F T

F T T

F F F


25/54

Negation

Negation of p: in symbols ~p

~p is false when p is true, ~p is true when p isfalse Example: p = "John is a programmer"

~p = "It is not true that John is a programmer"

p ~p

T F

F T


26/54

More compound statements

Let p, q, r be simple statements

We can form other compound statements,

such as (pq)^r

p(q^r)

(~p)(~q)

(pq)^(~r)

and many others


27/54

Example: truth table of (pq)^r

p q r (p q)^rT T T

T T F

T F T

T F F

F T T

F T FF F T

F F F


28/54

Section 1.3 Conditional propositions

and logical equivalence

A conditionalproposition is of the form

If p then q

In symbols: p q Example:

p = " John is a programmer"

q = " Mary is a lawyer "

p q = If John is a programmer then Mary is

a lawyer"


29/54

Truth table of p q

p q is true when both p and q are true

or when p is false (true by default or vacuously true)

p q p q

T T T

T F F

F T T

F F T


30/54

Hypothesis and conclusion

In a conditional proposition p q,

p is called the antecedentor hypothesis

q is called the consequentor conclusion

If "p then q" is considered logically the

same as "p only if q"


31/54

Necessary and sufficient

A necessarycondition is expressed by the

conclusion.

A sufficientcondition is expressed by the

hypothesis. Example:

If John is a programmerthen Mary is a lawyer"

Necessary condition: Mary is a lawyer Sufficient condition: John is a programmer


32/54

Logical equivalence

Two propositions are said to be logically

equivalentif their truth tables are identical.

Example: ~p q is logically equivalentto p q

p q ~p

q p q

T T T T

T F F F

F T T T

F F T T


33/54

Converse

The converseof p q is q p

These two propositions

are not logically equivalent

p q p q q p

T T T T

T F F T

F T T F

F F T T


34/54

Contrapositive

The contrapositive(or transposition) of the

proposition p q is ~q ~p.

They are logically equivalent.

p q p q ~q ~p

T T T T

T F F F

F T T T

F F T T


35/54

Biconditional proposition

The biconditional propersitionp if and only if qis defined in symbols as p q

p q is logically equivalent to (p q)^(q p)

p q p q (p q) ^ (q p)

T T T T

T F F F

F T F F

F F T T


36/54

De Morgans laws for logic

The following pairs of propositions are

logically equivalent:

~ (p q) and (~p)^(~q)

~ (p ^ q) and (~p) (~q)

S ti 1 4 A t d R l


37/54

Section 1.4 Arguments and Rules

of Inference

Deductive reasoning: the process of reaching a

conclusion q from a sequence of propositions p1,

p2, , pn.

A (deductive) argument is a sequence ofpropositions written as

The symbol is read therefore.

The propositions p1, p2, , pnare calledpremisesor hypothesis.

The proposition q that is logically obtained

through the process is called the conclusion.


38/54

Rules of inference (1)

1. Law ofdetachmentor

modus ponens p q

p

Therefore, q

2. Modus tollens

p q ~q

Therefore, ~p


39/54


3. Rule ofAddition

p

Therefore, p q

4. Rule ofsimplification

p ^ q

Therefore, p

5. Rule of conjunction

p

q

Therefore, p ^ q


40/54


6. Rule of hypothetical syllogism

p q

q r

Therefore, p r

7. Rule of disjunctive syllogism

p q

~p

Therefore, q


41/54

Section 1.5 Quantifiers

Apropositional functionP(x) is a statement

involving a variable x

For example: P(x): 2x is an even integer

x is an element of a set D

For example, x is an element of the set of integers

D is called the domainof P(x)


42/54

Domain of a propositional function

In the propositional function

P(x): 2x is an even integer,

the domain D of P(x) must be defined, for

instance D = {integers}.

D is the set where the x's come from.


43/54

For everyand for some

Most statements in mathematics and

computer science use terms such as for

everyand for some.

For example:

For everytriangle T, the sum of the angles of T

is 180 degrees.

For everyinteger n, n is less than p, for someprime number p.


44/54

Universal quantifier

One can write P(x) for everyx in a domain D

In symbols: x P(x)

is called the universal quantifier


45/54

Truth of as propositional function

The statement x P(x) is

True if P(x) is true for every x D

False if P(x) is not true for some x

D Example: Let P(n) be the propositional

function n2+ 2n is an odd integer

n D = {all integers}

P(n) is true only when n is an odd integer,

false if n is an even integer.


46/54

Existential quantifier

For somex D, P(x) is true if there exists

an element x in the domain D for which P(x) is

true. In symbols: x, P(x)

The symbol is called the existential

quantifier.


47/54

Counterexample

The universal statement x P(x) is false ifx D such that P(x) is false.

The value x that makes P(x) false is called acounterexampleto the statement x P(x). Example: P(x) = "every x is a prime number", for

every integer x.

But if x = 4 (an integer) this x is not a primernumber. Then 4 is a counterexample to P(x)being true.


48/54

Generalized De Morgans

laws for Logic If P(x) is a propositional function, then each

pair of propositions in a) and b) below have

the same truth values:a) ~(x P(x)) and x: ~P(x)

"It is not true that for every x, P(x) holds" is equivalentto "There exists an x for which P(x) is not true"

b) ~(x P(x)) and x: ~P(x)"It is not true that there exists an x for which P(x) istrue" is equivalent to "For all x, P(x) is not true"


49/54

Summary of propositional logic

In order to prove the

universally quantified

statement x P(x) is

true It is not enough to

show P(x) true for

some x D

You must show P(x) istrue for every x D

In order to prove the

universally quantified

statement x P(x) is

false It is enough to exhibit

some x D for which

P(x) is false

This x is called thecounterexampleto

the statement x P(x)

is true


50/54

Rules of inference for

quantified statements

1. Universal instantiation

xD, P(x) d D

Therefore P(d)

2. Universal generalization

P(d) for any d D Therefore x, P(x)

3. Existential instantiation

x D, P(x) Therefore P(d) for some

d D

4. Existential generalization

P(d) for some d D Therefore x, P(x)


51/54

Section 1.6 Nested Quantifiers

By definition, the statement xyP(x, y),

with domain of discourse XY, is true if, for

everyxX and everyyY, P(x, y)is true.The statement xyP(x, y)is false if there is

at least onexX and at least oneyY

such that P(x, y)is false.


52/54

xyP(x, y)

By definition, the statement xyP(x, y),with

domain of discourse XY, is true if, for every

xX, there is at least one yY for whichP(x, y)is true. The statement xyP(x, y)is

false if there is at least onexX such that

P(x, y)is falseforevery yY.


53/54

xyP(x, y)


domain of discourse XY, is true if there is at

least onexX such that P(x, y)is true forevery oneyY. The statement xyP(x, y)

is false if, for every xX, there is at least

oneyY such thatP(x, y)is false.


54/54

xyP(x, y)


domain of discourse XY, is true if there is at

least onexX and at least oneyY suchthat P(x, y)is true. The statement xyP(x,

y) is false if, for every xX and for every y

Y, P(x, y)is false.

csc331 week 1 topic a

Documents