Claudio Canuto - Mathematical Analysis I

Mathematical Analysis I — read online for free the complete book (whole text) full work

Below is the text of the book, divided by pages. System saving the place of the last page read, allows you to conveniently read the book "Mathematical Analysis I" online for free, without having to search again every time where you left off. Put a bookmark, and you can go to the page where you finished reading at any time.

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

1234567...148

Springer International Publishing Switzerland 2015

Claudio Canuto and Anita Tabacco Mathematical Analysis I UNITEXT 10.1007/978-3-319-12772-9_1

1. Basic notions

Claudio Canuto 1 and Anita Tabacco 1

(1)

Department of Mathematical Sciences, Politecnico di Torino, Torino, Italy

In this introductory chapter some mathematical notions are presented rapidly, which lie at the heart of the study of Mathematical Analysis. Most should already be known to the reader, perhaps in a more thorough form than in the following presentation. Other concepts may be completely new, instead. The treatise aims at fixing much of the notation and mathematical symbols frequently used in the sequel.

1.1 1.1 Sets

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:

• = set of natural numbers
• = set of integer numbers
• = set of rational numbers
• = set of real numbers
• = set of complex numbers.

The definition and main properties of these sets, apart from the last one, will be briefly recalled in .

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 . Prom the intuitive point of view it may be useful to represent subsets as bounded regions in the plane using the so-called Venn diagrams (see , left).

Figure 1.1.

Venn diagrams (left) and complement (right)

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

the order in which elements appear is not essential. This clearly restricts the use of such notation to subsets with few elements. More often the notation

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, i.e., the condition that is valid for the elements of the subset only, and not for other elements. For example, the subset A of natural numbers smaller or equal than 4 may be denoted

The expression p ( x ) = x 4 is an example of predicate , which we will return to in the following section.

The collection of all subsets of a given set X forms the power set of X , and is denoted by . Obviously . Among the subsets of X there is the empty set , the set containing no elements. It is usually denoted by the symbol , so . All other subsets of X are proper and non-empty.

Consider for instance X = {1, 2, 3} as ambient set. Then

Note that X contains 3 elements (it has cardinality 3), while has 8 = 23 elements, hence has cardinality 8. In general if a finite set (a set with a finite number of elements) has cardinality n , the power set of X has cardinality 2 n .

Starting from one or more subsets of X , one can define new subsets by means of set-theoretical operations. 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 (, right).

Sometimes, in order to underline that complements are taken with respect to the ambient space X , one uses the more precise notation . The following properties are immediate:

For example, if X = and A is the subset of even numbers (multiples of 2), then is the subset of odd numbers.

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

containing the elements of X that belong to both A and B , and union of A and B the subset

made of the elements that are either in A or in B (this is meant non-exclusively, so it includes elements of A B ), see .

Figure 1.2.

Intersection and union of sets

We recall some properties of these operations.

i)

Boolean properties :

ii)

commutative, associative and distributive properties :

iii)

De Morgan laws :

Next page

1234567...148

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

Similar books «Mathematical Analysis I»

Look at similar books to Mathematical Analysis I. We have selected literature similar in name and meaning in the hope of providing readers with more options to find new, interesting, not yet read works.

Gabriele Marcotti

Hail, Claudio!: The Man, the Manager, the Miracle

Claudio Hernández

The Beginnings of Stephen King

Susan Lewis

Claudio Monteverdi: A Research and Information Guide

Claudio Canuto

Mathematical Analysis II

Gazzola

Elements of Advanced Mathematical Analysis for Physics and Engineering

Cooke Roger

Mathematical Analysis II

César Pérez López

MATLAB Mathematical Analysis

Claudio Oliveira Egalon [Egalon

Mechanics: Experiments in Physics

Claudio Naranjo

The Enneagram of Society: Healing the Soul to Heal the World

Claudio Feser

When Execution Isn’t Enough Decoding Inspirational Leadership

Claudio Naranjo

Character and Neurosis: An Integrative View

Claudio Magris

Danube: A Sentimental Journey from the Source to the Black Sea

Discussion, reviews of the book Mathematical Analysis I and just readers' own opinions. Leave your comments, write what you think about the work, its meaning or the main characters. Specify what exactly you liked and what you didn't like, and why you think so.