Math Analysis I - Basic Notions

BASIC NOTIONS

SETS THEORY

Sets

A set is a list of elements which share a common characteristic. We shall denote sets

mainly by upper case letters X, Y, . . ., while for the members or elements of a set lower

case letters x, y, . . . will be used. When an element x is in the set X one writes x∈X (‘x

is an element of X’, or ‘the element x belongs to the set X’), otherwise the symbol x ∈ X

is used.

The majority of sets we shall consider are built starting from sets of numbers. Due to

their importance, the main sets of numbers deserve special symbols, namely:

IN s e t numbers

n a t u r a l

= o f

Z s e t integer numbers

o f

=

IOI s e t nationals numbers

o f

=

IR s e t r e e l numbers

= o f

¢ s e t numbers

complex

= o f

Let us fix a non-empty set X, considered as ambient set. A subset A of X is a set all of

whose elements belong to X; one writes A⊆X (‘A is contained, or included, in X’) if the

subset A is allowed to possibly coincide with X, and A⊂X (‘A is properly contained in X’)

in case A is a proper subset of X, that is, if it does not exhaust the whole X.

From the intuitive point of view, it may be useful to represent subsets as bounded

regions in the plane using the so-called Venn diagrams.

§ Eat

1. Math Analysis Basic Notions 1 of 74

A subset can be described by listing the elements of X which belong to it.

The order in which elements appear is not essential.

z}

{x,

A - y , . . .

More often the notation by property will be used (read ‘A is the subset of elements x of X

such that the condition p(x) holds’); p(x) denotes the characteristic property of the

elements of the subset. part}

part} {x

{x X I A-

A- X :

t e

{X

A-lo, a} tea}

1

A- IN

E

1,43, →

The collection of all subsets of a given set X forms the power set of X and is denoted by

P(X). Obviously X ∈ P(X). Among the subsets of X there is the empty set, the set

containing no elements. It is usually denoted by the symbol so ∈ P(X). All other

∅, ∅

subsets of X are proper and non-empty.

{I, 2,3} RX)

X- 8=23

X has has

cardinality = 3 , X}

{0, his},

13},

124,

P {in}, {2,3},

{i},

(X) - P 2"

X (X)

If has

condimeity

finite h a s

set n ,

a

1. Math Analysis Basic Notions 2 of 74

Operations w i t h sets

Starting from one or more subsets of X, one can define new subsets by means of

set-theoretical operations.

Compeement

The simplest operation consists in taking the complement: if A is a subset

of X, one defines the complement of A (in X) to be the subset made of all

elements of X not belonging to A. EA}

{xe Ix

(CA) K

-

Sometimes, in order to underline that complements are taken with respect to

G A

the ambient space X, one uses the more precise notation

The following properties are immediate: CCCCA))

- X

Ck) - A

CHI-f

For example, in X = N and A is the subset of even numbers (multiple of 2),

CCA)

then is the subset of all odd numbers. X-2k}

{X /

{Or io...}

A- A- IN

E

2 , 4 1 6 , oo, I}

{XEIN

(1,3, /

((A) a...} CCA) X - 2 k t

- 5 , 7, =

1. Math Analysis Basic Notions 3 of 74

Int e z section

Given the two subsets A and B of X, one defines the intersection of A and B

the subset containing the element of X that belong to both A and B

B}

{X X 1

M

A B A and

E x r E

E

=

€

¥i÷÷÷÷÷÷.

Mui o n

Given the two subsets A and B of X, one defines the union of A and B the subset

containing the element of X that are either in A or in B. B}

{X X 1

A V B A

E x x E

E

= o r

.€

€÷÷÷÷÷÷:..

1. Math Analysis Basic Notions 4 of 74

Properties

Booleenpropeztiere

A n - A

CCA) AUCCA) - X

Commutativeprol-pezties

AV

AB B - B U A

A BAA

=

Associetive.pro#peties-

CAhB)nc=AnCBnc)CAUB)UC=AUCBUc)

Distributirepr-operties

c) c)

A

( A h (AU

( B b c ) ( B n c )

YA U

B) FA A

B)

U c U

n c

Demongentrewse

((ARB) CLAN 431 CCA)

(AU B) MCCB)

C

= =

1. Math Analysis Basic Notions 5 of 74

Difference

The difference between a subset A and a subset B, sometimes called relative

complement of B in A, selects the elements of A that do not belong to B.

B}

{x 1

A 1 of

B A

c- x

=

Foo

Symmetric Difference

The symmetric difference of the subsets A and B picks out the elements

belonging either to A or B, but not to both.

B - A l

A- B) (BIA) BI

CA

CA BI

A V U A

I

= X IN

T.io#:..:.::..:i::::

E

e. g. {x x-2k}

/

A X

E

=

:::::::::::...

1. Math Analysis Basic Notions 6 of 74

LOGIC

Proposition

In Mathematical Logic a formula or proposition is a declarative sentence, or

statement, the truth or falsehood of which can be established. Thus within a

certain context a formula carries a truth value: True or False.

The truth value can be variously represented, for instance using the binary

value of a memory bit (1 or 0), or by the state of an electric circuit (open or

close).

Examples of formulas are: ‘7 is an odd number’ (True), ‘3 > 12’ (False), ‘Venus is

a star’ (False), et cetera. The statement ‘Milan is far from Rome’ is not a

formula, at least without further specifications on the notion of distance; in

this respect ‘Milan is farther from Rome than Turin’ is a proposition. We shall

indicate formulas by lower case letters p, q, r, . . ..

Logical. Connectives

New propositions can be built from old ones using logic operations expressed

by certain formal symbols, called logical connectives.

The concept of set can be understood only by using them.

Tabees

Tr u t h

A truth table is a mathematical tool used to show how the truth value of a

compound statement depends on the truth or falsity of the simple propositions

from which it's compounded.

P Q p

§T/...

f . . .

°mectia\

T

F . . .

F

F . . .

1. Math Analysis Basic Notions 7 of 74

Negation (not)

"7" o p e r a t o r

u n ve r y

-

7

§ p

p I

if 7

P P 2

2 7 3 3

=

= P,

Negation t h e

c o n ve r t values of @

F T

Conjunction "A" (end) bineryopez-eton

FF

p

a n e

p (T)

P-2 is e v e n

F

T F (T)

Q-4 is squeeze

a

F

F T

F F

F 5 = 8 - 5 K IN (F)

R E

Conjunction true

i s PA Q T PA S F

only b o t h d a r e =

P,

i f =

"V" ( o r )

D i s j u n c t i o n

binezyope-etnp-

i.EE#&ieYiitsi:ais.oee:

1. Math Analysis Basic Notions 8 of 74

"V. "

Exclusive Disjunction (Ron) bineryope-eton

÷±÷i÷÷o÷÷÷÷÷÷÷

"

" (if,

Implication implies)

then,

Tuesis

Hypothesis P

÷I¥µF

Q a

P CFI

P-2 odd

is (F)

K E N

6=5

Q K

=

T P t h e hypothesis

Q T

T

F a s

= verified

not

i s

l

I t only

false done i s

is i f ⇐

Logic "

Equivalence-Co-Implication " (if if)

and oney

⇐

p

ILIFFE

e

p

p ftp.isaetweeaonaeeyis

1. Math Analysis Basic Notions 9 of 74

Equivalence

Two propositions are considered equivalent if they present the same truth table.

In particular, a proposition made up by implication and logical equivalence as

connectives can be simplified by using basic connectives such as negation,

conjunction and disjunction. These can be called rules of interference.

P-7QisuotequiveleuttoQ.PL

÷t÷t÷t÷

p

P Q

Q a p

P Qisequiveleuttonp#

PfPyQfPVQ÷t¥÷::::

P a e

p T T

F T

T

F F T

P Q equivalent (contropositive

t o

is ' Q -p " "

÷t÷÷÷÷÷÷÷

T

1. Math Analysis Basic Notions 10 of 74

PAQisequiveleutt )

÷t÷t÷t÷t÷t÷÷t

'PV> nGpV7Q)

P Q

9 PA Q ' p Q

n F

Properties

Commutativity " y " ' " µ "

'✓ 11

The connectives Conjunction , Disjunction , Exclusive Disjunction

"Ey"

and Logical Equivalence (or co-implication) , satisfy commutative properties.

÷¥÷t÷t÷

P QAP

P Q

A

Q

1. Math Analysis Basic Notions 11 of 74

Particular propositions

" A "

Tautology

In logic, a tautology is a formula or proposition that is true in every

possible interpretation: "the ball is green, or the ball is not green", it is

either one or the other, it cannot be both and there are no other

possibilities.

A tautology is always true regardless of the truth value of the propositions

which compound the final formula.

" 1 "

Contradiction

A contradiction consists of a logical incompatibility between two or more

propositions: “the ball is green and not green at the same time”.

Therefore, a contradiction is always false.

(P Q)

Q t a u t e r

• Q) 'Q)

(p (p

1 Proposition

- A

• P ' P - A

f # CONTRADICTION2

•

1. Math Analysis Basic Notions 12 of 74

Predicate

A predicate is an assertion or property p(x, y, . . .) that depends upon one or more

variables x, y . . . belonging to suitable sets, and which becomes a proposition

(hence true or false) whenever the variables are fixed.

"risodd"

pix) -

if p(x)

X - l o t r u e

i s

p(x) FALSE

X - 7

if is

Observe that the aforementioned logic operations can be applied to predicates

as well, and give rise to new predicates (e.g., ￢p(x), p(x) V q(x)).

This fact, by the way, establishes a precise relation among the essential connectives

" T ", "A, "V", and set operations: complement, intersection, union.

"V",

In fact, recalling the definition A = {x X | p(x)} of subset of a given set X, the

E

‘characteristic property’ p(x) of the elements of A is nothing else but a predicate,

which is true precisely for the elements of A. part} {0,214,6...}

{XEXI

p w - "xiseveu" A -

A-

The complement is thus obtained by negating the characteristic property:

CCA) {XEX

(CA) l a p e l }

-

Reiterating the same process for proper logical connectives, it becomes possible to

express set operations: intersection, union, difference, symmetric difference,

by describing the corresponding characteristic properties.

{reXIparthgal}

An B-

A:{xEX Ipa)} {xeXlpartracx)}

A- U B -

{REX law}

B - 17941}

{XEXI

A , B - par)

A B - {xexlpcxlv.ae)}

A-

1. Math Analysis Basic Notions 13 of 74

Quantifiers

Given a predicate p(x), with the variable x belonging to a certain set X, one is

naturally lead to ask whether or not p(x) is true for all elements x, or if there exists

at least one element x making p(x) true.

When posing such questions we are actually considering the following formulas:

Universal q u a n t i t i e s

Fx "for

X, pal holds"

E pix)

aeese,

V x IN,

E

e . xxo

g.

Existential quantifier

Fx X, "there

E pix) least

exists a t a ,

o n e

t h a t h o l d s "

such part

I x EIN, 5 7

X

e. g.

11

F! X, there

part

E e x i s t s

x unione a ,

a

and o n l y,

one o n e

t h a t h o l d s "

such part

F! I R ,

E 3 = 7

e . X t

g.

By putting a quantifier in front of a predicate it is transformed into a formula,

whose truth value may be then determined.

However the effect of negation on a quantified predicate must be handled

with attention. Suppose for instance x indicates the generic student of the

Polytechnic, and let p(x) = ‘x is an Italian citizen’. The formula ‘∀x, p(x)’ (‘every

student of the Polytechnic has Italian citizenship’) is false. Therefore its

negation ‘￢(∀x, p(x))’ is true, but beware: the latter does not state that all

students are foreign, rather that ‘there is at least one student who is not

Italian’. (tx ⇐

pix)) Ex

E X , -par)

X ,

E

> (Fx ⇐

K , (x)) X ,

f r

E par t

- E a

p

1. Math Analysis Basic Notions 14 of 74

Properties

If a predicate depends upon two or more arguments, each of them may be

quantified. Yet the order in which the quantifiers are written can be

essential.

Namely, two quantifiers of the same type (either universal or existential) can

be swapped without modifying the truth value of the formula;

in other terms:

f x t y, itty#,

⇐

p l a y ) play)

Fy,

Fx F i x ,

⇐ play)

p l a y ) >

On the contrary, exchanging the places of different quantifiers usually leads to

different formulas, so one should be very careful when ordering quantifiers.

"x y'

y)

(x, ±

P =

Fy

Fx I R , y)

( x ,

E t r u e

p

IR,

Fxfy Y)

E p i , FA L S E

Fxty PA, Y)

I R , FALSE

E

FAY par,'ll

R ,

E T H E

1. Math Analysis Basic Notions 15 of 74

SETS NUMBERS

O F

IN-nonnegative

Set

This set has the numbers 0, 1, 2,... as elements.

The operations of sum and product are defined on N and enjoy the well-known

commutative, associative and distributive properties.

Operations II;

Sum(m,u) → m= t . . . . t e ,

t ez

t

a n e ,

o o

=

D Commutativity IN

t

t h E

m u t

t m m, n

2) A s s o c i a t e

a)

( m t (ut tm,

k) IN

E

R - m U

t

t n ,

3) N e u t r u e n t IN

t m E

m t o O f m - m

=

Prwowctlm.nl#m.n-

D Commutativity IN

t E

m . w e n . m,

m n

2) A s s o c i a t e

a)

( m . (n. tm,

R) IN

E

K - m . U

. n ,

3) N e u t r o n IN

F m E

m - s - s . m

m =

4) Distributivity IN

Fm.

(mta) K E

R

k k m u n ,

t

=

1. Math Analysis Basic Notions 16 of 74

Properties

Notation Nt

We shall indicate by the set of natural numbers different from 0

{o}

Nt l

IN

=

A natural number n is usually represented in base 10 by the following expansion: 0

¥ Cielo"

Z, Cielo" l o " '. . .

n = -A t c a , t co

t

c, lo

hell IN

E decimal

o f m e

C, 5

where digits

= o n e o f n

(Cp 6)

C u . ,

n " c '

= ' (IO)

to BASE

(Cp 6 )

C

n n - i - r - c '

= 2 - a (2)

BINARY BASE

Representation

Natural numbers can also be represented geometrically as points on a

straight line.

For this it is sufficient to fix a first point O on the line, called origin, and

associate it to the number 0, and then choose another point P different from

O, associated to the number 1. The direction of the line going from O to P is

called positive direction, while the length of the segment OP is taken as unit

for measurements.

By marking multiples of OP on the line in the positive direction we obtain the

points associated to the natural numbers. to

0 1 9 ¥

2 3

L

O

1. Math Analysis Basic Notions 17 of 74

Principle

Induction

The Principle of Mathematical Induction represents a useful rule for proving

properties that hold for any integer number n, or possibly from a certain

integer n0 ∈ N onwards. In order to use induction principle, a first heuristic

deduction or a very strong conjecture is needed.

Let n0 ≥0 be an integer and denote by P(n) a predicate defined for every

integer n≥n0. (n)

fu

% %

no no, p

o

Suppose the following conditions hold:

e) (no) true

p INDUCTION

is BASIS I N D U CT I O N

ii) Fu 1)

pcu)

% par t true

n o , - A is step

tu IN

theorem

pen)

p(u) true-a i n

is is

% no, e

Proof (byington) Cn-l) t u

(n-2)

u! EIN

(n) - 2 . s

u

=

=

p . . .

a) o ! b y definition

= L

iidcutil! (htt).. s - n ! (htt)

Cu-z)....

n ( n - 1 )

-

Theorem Gauss

o f

Sumeelwumbersfnomitoi

5

3 4

2

I

I, to to

to

to

t o 99 96

98 97

100 L O L L O L

L O L L O L L O L 50

lol.

S o m e =

5050

=

1. Math Analysis Basic Notions 18 of 74

Generalisation Gauss

of

Sumofalluumbersjwnstont

1) w i s e 5

3 4

2

I

| to to

to

to

to

r n - I n - 3

n-2

u u - u I

(htt)

h t t

h t t sum--

h t t

h t t

h t t

1)n i add computatiousftotnenti)

the

t o

o 3 4

2

0 I

| to to

to

to

to

r n - I n - 3

n-2

u h - h

sum-u.tn#Pnooj/

h

h

h

h

h

byiuow-tiou)

u 412¥

[pans

Pln) e n o -06¥-o

-Egan,

e) pro) - h a t h

Iopa,

ii)p(a) = 2

"I§pCn)=ChH){nt#=(utµ)

inJpluti) -

Induction:¥§pCn) =h(nz#turtle

-I£p(a) t u t ,

=n(nt¥)=(htY{u¥ f u EIN, pal

1. Math Analysis Basic Notions 19 of 74

Statis