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Chapter I MATH FUNDAMENTALS I.1 Sets, Numbers, Coordinates, Functions August 16, 2020

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II..11 SSEETTSS,, NNUUMMBBEERRSS,, CCOOOORRDDIINNAATTEESS,, FFUUNNCCTTIIOONNSS Objectives: After the completion of this section the student should recall

- the definition of sets and operations associated with them

- the definition of sets of numbers

- the basic coordinate systems in Euclidian space

- the definition and classification of functions

Contents: 1. Sets

2. Operations with sets 3. Proofs

4. Numbers

5. Constants and variables

6. Coordinates

7. Functions

8. Review questions and exercises

Chapter I MATH FUNDAMENTALS I.1 Sets, Numbers, Coordinates, Functions August 16, 2020 4

1.1 SETS, NUMBERS, COORDINATES, FUNCTIONS In this section we recall the basics of mathematical language and notations. 1. SETS: The sets are collections of some objects. The theory of sets is a convenient and

universal form of description and operations with the different sets. We consider mostly the sets of numbers which can be the sets of natural and real numbers or intervals, the domains of the functions, the sets of all solutions of the algebraic equations, and the sets of functions which can represent the solutions of the differential and integral equations, the sets which form the vector spaces, etc. The theory of sets has much in common with mathematical logic and probability theory. Here we consider just some aspects of ‘naïve’ set theory rather than ‘axiomatic’ set theory.

Definition 1: We assume a set to be a well-defined collection of objects. These objects are said to be the elements of the set.

That the sets are well-defined means that for any object there can be only one of two possible cases regarding the given set: either this object belongs to a given set, or this object does not belong to a given set.

Capital letters A, B, C, … , X, Y, Z will be used for designation of the sets; and lower-case letters a, b, c, … , x, y, z will be used for designation of the elements of the sets.

For visual illustration of the sets, the symbolic closed circular regions will be used (the so called Venn diagrams), and points will be used for graphical representation of the elements of the sets. For example, the set X with the element x can be shown as

To show that an element a belongs to a set A, we will use the notation

a A∈

If b does not belong to a set B, then we will write

b B∉

There are two basic ways to describe the elements of the sets:

1) The set can be described by a simple ’listing’ of its elements. We will use a notation with braces of the kind

{ }A a,b,c,=

2) In the ‘tabular’ method, to describe some properties of the elements, the symbol “|” will be used in a sense “such as”. For example, if set A consists of all elements, which simultaneously belong to the sets B and C, we can write

{ }A a | a B and a C= ∈ ∈

An empty set is a set which does not posses any elements. A symbol ∅ is used for designation of the empty set.

Any part of a set is called a subset: if b B∈ then b A∈ .

Any subset is a set itself.

X

x

A

a

B

b

A

B

B is a subset of A

{ }A a | a B and a C = ∈ ∈

symbol " |" means "such that"


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2. OPERATIONS WITH SETS (algebra of sets): equality A B= Two sets A and B are said to be equal if they consist of the same elements.

Two conclusions can be made from the statement A B= : 1) if a A∈ then a B∈

2) if b B∈ then b A∈ To prove that two given sets are equal, we need to check the validity of these two statements. This definition yields that repeated listing of some element in the set or order of listing do not change the set. For example, the sets { }1,3 , { }3,1 , { }3,3,1 and { }1,1,3 are equal. That can be confirmed by checking the statements 1) and 2). If sets A and B are not equal, we write that A B≠ .

inclusion B A⊂ Set A contains set B or, what is equivalent, set B is contained in set A.

This means that all elements of the set B also belong to the set A. Only one conclusion can be made from this statement:

if b B∈ then b A∈

If also set B contains set A, then both statements 1) and 2) are satisfied, and, therefore, the sets A and B are equal. This constitutes an important axiom in set theory (axiom of extensionality): if for sets A and B, A B⊃ and B A⊃ , then A B= . In this case, A and B are said to be improper subsets of each other. If there exists at least one element in set A such that it does not belong to set B, then set B is called to be a proper subset of set A; and inclusion of B in A is called to be a strict inclusion. The standard notation (DIN 5473), which is widely used, refers to the case of a strict inclusion:

B A⊂ that means that B A⊂ and A B≠ . For improper inclusion, the notations A B⊆ and A B⊇ also are used.

union A B∪ The union of two sets A and B is a set which consists of all elements of set A

and set B. The conclusions which can be made about the elements of these sets:

if a A∈ , then a A B∈ ∪ if b B∈ , then b A B∈ ∪ if c A B∈ ∪ , then c A∈ and/or c B∈

Symbolically, the definition of the union of two sets is given by an expression:

{ }A B a | a A and / or a B∪ = ∈ ∈ The operation union is reflective: A B B A∪ = ∪ intersection A B∩ The intersection of two sets A and B is a set which consists of all elements

which simultaneously belong to set A and set B: { }A B c | c A and c B∩ = ∈ ∈ The conclusions about the intersection: if c A B∈ ∩ , then c A∈ and c B∈ and, consequently, if c A∈ and c B∈ , then c A B∈ ∩ The operation intersection is also reflective: A B B A∩ = ∩

A

B

A B=

A

BA B∩

A

B

A B∪


difference A\ B The difference of two sets A and B is a set which consists of all elements of set A that do not belong to set B (reads A without B):

{ }A\ B c | c A and c B= ∈ ∉

The conclusion about difference: if c A\ B∈ , then c A∈ and c B∉ if c A∈ and c B∉ , then c A\ B∈ Difference is not a reflective operation, in general: A\ B B\ A≠ complement cA If set A is a subset of set B, then the difference B\ A is called a complement of

set A with respect to set B:

{ }cA B\ A c | c B and c A= = ∈ ∉

Some properties of the complement of set A with respect to set B: cB = ∅

( )ccA A= cA A B∪ = cA A∩ =∅ c B∅ = product A B× A product (or Cartesian product) of two sets A and B is a set consisting of all

ordered pairs ( )a,b where a A∈ and b B∈ :

( ){ }A B a,b | a A,b B× = ∈ ∈ Cartesian product is not reflective, in general,

A B B A× ≠ ×

Cartesian products of higher orders are received by consecutive application of Cartesian product to more than two sets:

( ){ }A B C a,b,c | a A,b B,c C× × = ∈ ∈ ∈ and so on. We will denote the Cartesian product of set A with itself by the usual notation

2A A A= × and, in general, n

n times

A A A A= × × ×

Properties of set operations: 1) A B B A∪ = ∪ commutative law

A B B A∩ = ∩ 2) ( ) ( )A B C A B C∪ ∪ = ∪ ∪ associative law

( ) ( )A B C A B C∩ ∩ = ∩ ∩ 3) ( ) ( ) ( )A B C A B A C∩ ∪ = ∩ ∪ ∩ distributive law ( ) ( ) ( )A B C A B A C∪ ∩ = ∪ ∩ ∪ 4) ( ) ( ) ( )A\ B C A\ B A\ C∪ = ∩ De Morgan’s rules

( ) ( ) ( )A\ B C A\ B A\ C∩ = ∪

A

BA\ B

A

B

a

A B×

b

( )a,b

B

AcA


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3. PROOFS: It is important to know how statements in set theory are proved. Here, we provide some examples of proof of statements in set theory.

Example 1 (direct proof)

Prove the commutative law for intersection: A B B A∩ = ∩

Proof: To prove that two sets are equal we need to show the validity of two statements:

1) A B B A∩ ⊂ ∩

Let a A B∈ ∩ , then, according to the definition of intersection, a A∈ and a B∈ , this also means, that a B∈ and a A∈ (in mathematical logic, the statement x and y is equivalent to the statement y and x). Therefore, using the definition of intersection again, we have that a B A∈ ∩ . So, if a A B∈ ∩ , then a B A∈ ∩ . That means that A B∩ is a subset of B A∩ , or, in other words, that A B B A∩ ⊂ ∩ .

2) A B B A∩ ⊃ ∩

the proof of this part is completely similar to the previous one.

From 1) and 2) and the axiom of extensionality, it follows that

A B B A∩ = ∩ q.e.d. ■

Example 2 (proof by contradiction)

Prove that A∅ ⊂ for any set A (empty set is a subset of any set)

Proof: Assume that the statement we have to prove is not true, i.e. that the empty set ∅ is not a subset of any set; it means that there exists at least one set B such that ∅ is not a subset of B ; it means that there exists an element a∈∅ such that a B∉ ; but the empty set ∅ contains no elements, therefore, what we get contradicts the definition of the empty set; Therefore, our assumption that ∅ is not a subset of any set is not true; that means that the opposite statement is true:

∅ is a subset of any set. q.e.d. ■


4. NUMBERS: What we in the previous sections were abstract sets with elements symbolically denoted by letters. Here, we consider examples of the sets of common numbers, though we are not going to give strict definitions of these sets.

natural numbers = set of natural numbers, { }1,2,3,...= ,

numbers which appeared in counting and can be used in indexing ( 1 2 3a ,a ,a ,... ). integers = set of integers, { }..., 3, 2, 1,0,1,2,3,...= − − − .

rational numbers = set of rational numbers, a a , bb

= ∈ ∈

.

real numbers = set of real numbers, set of all numbers which are represented by the

positive or negative infinite decimals such as 35.9617… or –1.2593... Remarkable real numbers: 2 1.4142...= , e 2.7818..= ., 3.1392...π = .

The set ( ){ }2 = x,y x, y∈ represents the Euclidian plane and the set

( ){ }3 = x,y,z x, y,z∈ represents the 3-dimensional Euclidian space.

complex numbers = set of complex numbers, { }2a ib a,b , i 1 = + ∈ = − . Field of Numbers The algebraic structure of the sets of numbers is formulated in the definition of

the abstract algebraic notion of a field.

Field is a set F with two binary operations: addition ( + ) and multiplication ( ⋅) for which the following axioms are satisfied: 1) if x, y F∈ , then x y F+ ∈ (closure for addition)

2) if x, y F∈ , then x y F⋅ ∈ (closure for multiplication

3) x y y x+ = + for any x, y F∈ (commutative law)

x y y x⋅ = ⋅ for any x, y F∈ (commutative law)

4) ( ) ( )x y z x y z+ + = + + for any x, y,z F∈ (associative law) 5) ( ) ( )x y z x y z⋅ ⋅ = ⋅ ⋅ for any x, y,z F∈ (associative law) 6) ( )x y z x y x z⋅ + = ⋅ + ⋅ for any x, y,z F∈ (distributive law) 7) there exists a unique element 0 F∈ such that

x 0 0 x x+ = + = for any x F∈ (zero element) 8) there exists a unique element 1 F∈ such that

x 1 1 x x⋅ = ⋅ = for any x F∈ (unit element) 9) for any x F∈ there exists a unique number y F∈ such that

x y y x 0+ = + = (negative of x)

10) for any x F∈ ( x 0≠ ) there exists a unique number y F∈ such that

x y y x 1⋅ = ⋅ = (inverse of x)

The set of rational numbers , the set of real numbers and the set of complex numbers are fields. All of them posess the abovementioned properties.


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For the negative of x, we usually use the notation x− , and for the inverse of x,

we use the notation 1x

. The negative and the inverse allow us to introduce two

new operations with the elements of the field: subtraction and division.

Previously introduced sets were just arbitrary collections of abstract objects (although we considered only the objects of a mathematical nature). Now, the fields just introduced are sets in which mathematical operations are applied, and they also satisfy the determined conditions. The most significant is that fields are complete sets of the objects with the given properties – they include all elements necessary for performing the introduced operations.

Algebraic Rules Recall the main algebraic rules for real numbers . Let x, y,z,u,v∈ , then a) x z y z+ = + ⇒ x y=

x z y z⋅ = ⋅ , z 0≠ ⇒ x y=

b) x 0 0 x 0⋅ = ⋅ =

c) ( )x 1 x− = − ⋅

d) ( ) ( )x y x y− ⋅ − = ⋅

e) x y 0⋅ = ⇒ x 0= or y 0=

f) x u x v y uy v y v

⋅ + ⋅+ =

⋅

g) x u x uy v y v

⋅⋅ =

⋅

h)

xx v x vy

u y u y vv

⋅= ⋅ =

⋅

Powers Let x, y∈ and m,n∈ , then

a) nn times

x x x x⋅ ⋅ ⋅ ⋅ =

b) n n1xx

− =

c) 0x 1= , x 0≠

d) m n m nx x x +⋅ =

e) m

m nn

x xx

−=

f) ( )nm mnx x=

g) ( )n n nxy x y=

h) n n

n

x xy y

=


5. CONSTANTS AND VARIABLES

Two words “constant” and “variable” are used for representation of the elements of a set of numbers:

constant = particular element of a set;

variable = a symbol representing the elements from some set, which is called a range of a variable.

For example, some fixed element a from a set A is a constant.

The symbol x which can assume designation of all elements of the set A is a variable. We already used variables to illustrate the definitions of operations with sets. The most typical usage of a variable in set theory looks like:

for any x A∈

which means “for any element x in set A”.

symbol “=” The symbol “=” (equal) will be used in two different senses:

identity a b= means that constants a and b represent the same element;

equation x b= means that there exists an element b which can be used for replacement of variable x. In this case, we can say that variable x assumes the value b , or that the solution of this equation is b .

Properties of the equal sign:

1) x x= reflexivity 2) if x y= , then y x= symmetry 3) if x y= and y z= , then x z= transitivity

Equal is a particular case of a more general equivalence relation.


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6. COORDINATES Cartesian Coordinates We already adopted the notation for the set of all real numbers. The

geometrical figure associated with is a line. We fix a point 0 (which we call the origin) in the line l, then we choose the unit of length as a distance between origin and some other

point (which we denote 1) to the right of the origin. The real number 0 corresponds to the origin, and the real number 1 corresponds to the point called 1.

Then any other point a to the right of the origin corresponds to a positive real number which quantitatively is equal to the length of the segment between 0 and a in terms of the chosen unit length. The points to the left of the origin correspond to negative real numbers in the same manner. As a result, we have that for any real number a∈ , there exists a unique point on the line; and for any point on the line there corresponds a unique real number. Therefore, we can establish a one-to-one function from l into . This function which we denote by x : l → are Cartesian coordinates of the points on the line – each point on the line has a corresponding numerical value. The line l we call the real line, the direction to the right of the origin is a positive direction of the line which is denoted by an arrow. In terms of Cartesian coordinates, the line is said to be an x-axis.

The real numbers on the x-axis are ordered: if a and b are two distinct points on the real line, and point a is to the left of the point b, then a open;

[ ) { }b, x x b∞ = ≥ half-open. Here, we introduce the symbols ∞ and -∞ which do not correspond to any point

on the real line or to any real number.

2 Now, consider the Cartesian product ( ){ }2 a,b a,b= × = ∈ .

The geometrical figure associated with the set 2 is a plane.

Any point in the plane can be put in a one-to-one correspondence with the set 2 by introducing Cartesian coordinates. The figure shows an orthogonal

Cartesian coordinate system. Lines x and y are called the x-axis and y-axis, correspondingly. The value x is called an abscissa, and the value y is called an ordinate. In the coordinates ( )a,b , the abscissa value is always given first. We will use the orientation of the axis always as shown in the Figure (right-handed system).

l

1


3 The set of all triplets of real numbers ( ){ }3 a,b,c a,b,c= ∈ is associated

with 3-dimensional space. This association can be established by the orthogonal Cartesian coordinate system with three axes: abscissa x, ordinate y, and applicate z, which for a right-handed system has a form shown in the Figure.

The idea of a Cartesian coordinate system seems to be very simple, but the

discovery of this system took a long time. Though the origin of mathematical science reaches deep into the centuries B.C., the introduction of Cartesian coordinates was made by the French mathematician and philosopher Rene Descartes (Latin name Cartesian) only in the 17th century. It was, probably, the greatest revolution in mathematics since the achievements of Greek mathematicians.

We use Cartesian coordinates to describe the position of a point in space. There

are also other coordinate systems which can be used for the same purpose. In some cases, they are more convenient and simple than Cartesian coordinates. Here, we list these systems and show their relation to the Cartesian coordinate system.

polar coordinates in 2 (plane polar-coordinate system)

For polar coordinates in plane, we need a reference direction. It is convenient for further conversion, to choose the positive direction of the x-axis 0x as the reference direction. Then an arbitrary point in the plane P is associated with the ordered pair ( )r,θ where r is the distance between point P and the origin 0, and θ is a principal angle measured in radians in a counterclockwise direction between the reference direction 0x and the direction of 0P.

Polar coordinates of point P: ( )r,θ , where

r is the radius; r 0≥ (r

= >= = < + <


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cylindrical coordinates in 3 (a coordinate system which is of importance for problems with rotational symmetry about one of the axis)

Cylindrical coordinates describe the position of the point P in space by the triple

of numbers ( )r, ,zθ , where r is the radius of cylinder (distance between the point and the z-axis),

r 0≥

θ is the azimuth, angle between the positive direction of the x-axis and the direction of 0P′ , where the point P′ is the orthogonal projection of the point P onto the xy-plane; 0 2θ π≤ <

z is the Cartesian z coordinate of the point P cylindrical to Cartesian Cartesian to cylindrical

x r cosθ= 2 2 2r x y= +

y r sinθ= ytanx

θ =

z z= z z=

spherical coordinates in 3 (a coordinate system which is of importance for problems with rotational symmetry about the origin)

Spherical coordinates describe the position of the point P in space by the triple

of numbers ( )r, ,φ θ , where r is the radius, a distance between the point P and the origin 0 ; r 0≥

θ is the latitude, the angle between the direction of 0P and the positive direction of the z-axis; 0 θ π≤ ≤

φ is the longitude, the angle between the positive direction the of x-axis and the direction of 0P′ , where the point P′ is the orthogonal projection of the point P onto the xy-plane; 0 2φ π≤ <

spherical to Cartesian Cartesian to spherical x r sin cosθ φ= 2 2 2 2r x y z= + +

y r sin sinθ φ= 2 2 2

z zcosr x y z

θ = =+ +

z r cosθ= ytanx

φ =

spherical to cylindrical ( ), ,zρ θ cylindrical to spherical ( )r, ,φ θ

r sinρ θ= 2 2r zρ = +

θ φ= φ θ=

z r cosθ= arctanzρθ =

P′

P

P

P′


7. FUNCTIONS Definition 2 Let A and B be two nonempty sets (which can be distinct or the same). A function f of set A into set B is a mapping which for every element of set A gives in correspondence precisely one element in set B .

Notation for a function:

f : A B→ ( )y f x= , x A∈ , y B∈

Here, x is an independent variable, and y is a dependent variable – the value of the function f at x. We also call y an image of x under the mapping f. Set A is said to be a domain of function f, and the set B is the co-domain. The set of all images R is called the range of function f:

( ){ }R y | y f x ,x A= = ∈

This definition of a function is traditional in calculus. The other (more formal) definition: Let A and B be two nonempty sets. Function f : A B→ is a set of ordered pairs ( ){ }f x, y= such that: 1) for any x A∈ , there exists y B∈ such that ( )x, y f∈ ; 2) if ( )x, y f∈ and ( )u,v f∈ , and if x u= , then y v= . This definition also provides a basis for the graph of the function ( )y f x= which is the set of points with the coordinates ( )( ){ }x, f x x A∈ .

The real valued function ( )y f x= of the real variable is usually defined by the equation without indication of the domain, assuming that the domain is the largest subset of for which the function ( )y f x= is defined. For example, if the function is given by the equation

2

1y1 x

=−

then its domain is ( )A 1,1= − and its range is [ )R 1,= ∞ . The graph of this function is shown on the left.

constant function A function which for all x A∈ assumes the same value a B∈ . A constant function is given by the equation: y a= inverse function Let f be a function from A into B, and let Y B⊂ be a range of the function f. If

there exists a function g from Y into A, such that ( )( )g f x x= for all x A∈ , then the function f is said to be invertible and the function g is called the inverse of the function f. Notation 1f − is used for the inverse function:

( )1x f y−=

x

domain

y

( )y f x=

A

co domain−

B

range

( )( ) ( ){ } x, f x x A, f x B∈ ∈

x y

( )y f x=

A B

range

( )1x f y−=

Y

2

1y1 x

=−


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one-to-one function The function f : A B→ is called one-to-one if from x y≠ follows ( ) ( )f x f y≠ for all x, y A∈ .

Recall the horizontal line test for a one-to-one function.

Theorem 1 The function is invertible if and only if it is one-to-one.

Proof: 1) ⇒ (if a function is invertible, then it is one-to-one) Let x, y A∈ be such that x y≠ .

Suppose that ( ) ( )f x f y= , then ( )( ) ( )( )1 1f f x f f y− −=

given that the inverse function exists. And, consequently, from the definition of the inverse function, we have that x y= that contradicts the condition that x y≠ , therefore, our assumption that ( ) ( )f x f y= is wrong, and we have that ( ) ( )f x f y≠

when x y≠ , that means that f is one-to-one. 2) ⇐ (if the function is one-to-one, then it is invertible)

■ monotonic functions The monotonic functions are the functions of one of the following types: a) ( )f x is an increasing function on the interval ( )a,b if ( ) ( )1 2f x f x< for all ( )1 2x ,x a,b∈ such that 1 2x x< b) ( )f x is a non-decreasing function on the interval ( )a,b if ( ) ( )1 2f x f x≤ for all ( )1 2x ,x a,b∈ such that 1 2x x<

c) ( )f x is a decreasing function on the interval ( )a,b if ( ) ( )1 2f x f x> for all ( )1 2x ,x a,b∈ such that 1 2x x< d) ( )f x is a non-increasing function on the interval ( )a,b if

( ) ( )1 2f x f x≥ for all ( )1 2x ,x a,b∈ such that 1 2x x<

Theorem 2 If the function ( )f x is either increasing or decreasing on ( )a,b , then ( )f x is invertible on ( )a,b .

composition Let R be the range of the function ( )f x , x A∈ and let

the function ( )g x , x R∈ be defined on R . Then the composition of f and g is the function

( )g f g f x= Example 3 Let ( ) 2f x x= and ( )g x x 1= + Then ( ) 2g f g f x x 1= = + Note, that the composition ( ) ( )2f g f g x x 1= = + Therefore, in general, f g g f≠ .

xf

A B

( )g f g f x=

R

g

x

y

f

A B

( )f x

( )f y


8. REVIEW QUESTIONS: 1) What are sets?

2) What are two basic ways to describe the elements of sets?

3) Recall the operations with sets.

4) Recall the sets of numbers?

5) Which sets of numbers are the subsets of other sets of numbers?

6) Why the sets of natural numbers and the set of integers are not the fields?

7) What is the difference between constants and variables?

8) What is the difference between the identities and equations?

9) Recall the definition of major coordinate systems.

10) Recall the definition of function.

11) What monotonic functions are one-to-one?

12) If function satisfies the horizontal line test, is it invertable?

EXERCISES: 1) Prove the following properties of the operations with sets: a) A B B A∪ = ∪ commutative law for union b) A B B A∩ = ∩ commutative law for intersection 2) Prove or disapprove the following statement:

if A B A∩ = , then A B B∪ =

3) Prove that if from ( ) ( )f x f y= follows that x y= , then function f is invertible.

4) Prove or disapprove the following statement:

Let x, y∈ . If x xy 1+ + is even, then x is odd. 5) Prove that if a function is one-to-one, then it is invertible.

I.1 SETS, NUMBERS, COORDINATES, FUNCTIONSObjectives: After the completion of this section the student should recall2. Operations with sets3. Proofs4. Numbers5. Constants and variables6. Coordinates7. Functions1.1 SETS, NUMBERS, COORDINATES, FUNCTIONS1. SETS: The sets are collections of some objects. The theory of sets is a convenient and universal form of description and operations with the different sets. We consider mostly the sets of numbers which can be the sets of natural and real numbers ...

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