Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 3 I. SETS, NUMERS, COORDINTES, FUNCTIONS Objetives: fter the ompletion of this setion the student should reall - the definition of sets and operations assoiated with them - the definition of sets of numbers - the basi oordinate sstems in Eulidian spae - the definition and lassifiation of funtions Contents:. Sets. Operations with sets 3. Proofs 4. Numbers 5. Constants and variables 6. Coordinates 7. Funtions 8. Review questions and eerises
4 Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08. SETS, NUMERS, COORDINTES, FUNCTIONS In this setion we reall the basis of mathematial language and notations.. SETS: The sets are olletions of some objets. The theor of sets is a onvenient and universal form of desription and operations with the different sets. We onsider mostl the sets of numbers whih an be the sets of natural and real numbers or intervals, the domains of the funtions, the sets of all solutions of the algebrai equations, and the sets of funtions whih an represent the solutions of the differential and integral equations, the sets whih form the vetor spaes, et. The theor of sets has muh in ommon with mathematial logi and probabilit theor. Here we onsider just some aspets of naïve set theor rather than aiomati set theor. Definition : We assume a set to be a well-defined olletion of objets. These objets are said to be the elements of the set. That the sets are well-defined means that for an objet there an be onl one of two possible ases regarding the given set: either this objet belongs to a given set, or this objet does not belong to a given set. Capital letters,, C,, X, Y, Z will be used for designation of the sets; and lower-ase letters a, b,,,,, z will be used for designation of the elements of the sets. For visual illustration of the sets, the smboli losed irular regions will be used (the so alled Venn diagrams), and points will be used for graphial representation of the elements of the sets. For eample, the set X with the element an be shown as X a b To show that an element a belongs to a set, we will use the notation a If b does not belong to a set, then we will write b There are two basi was to desribe the elements of the sets: ) The set an be desribed b a simple listing of its elements. We will use a notation with braes of the kind { a,b,, } = smbol " " means "suh that" = { a a and a C } is a subset of ) In the tabular method, to desribe some properties of the elements, the smbol will be used in a sense suh as. For eample, if set onsists of all elements, whih simultaneousl belong to the sets and C, we an write = { a a and a C} n empt set is a set whih does not posses an elements. smbol is used for designation of the empt set. n part of a set is alled a subset: if b then b. n subset is a set itself.
Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 5. OPERTIONS WITH SETS (algebra of sets): equalit = Two sets and are said to be equal if the onsist of the same elements. Two onlusions an be made from the statement = : ) if a then a ) if b then b = To prove that two given sets are equal, we need to hek the validit of these two statements. This definition ields that repeated listing of some element in the set or order of 3,3, and inlusion listing do not hange the set. For eample, the sets {,3 }, { 3, }, { } {,,3 } are equal. That an be onfirmed b heking the statements ) and ). If sets and are not equal, we write that. Set ontains set or, what is equivalent, set is ontained in set. This means that all elements of the set also belong to the set. Onl one onlusion an be made from this statement: if b then b If also set ontains set, then both statements ) and ) are satisfied, and, therefore, the sets and are equal. This onstitutes an important aiom in set theor (aiom of etensionalit): if for sets and, and, then =. In this ase, and are said to be improper subsets of eah other. If there eists at least one element in set suh that it does not belong to set, then set is alled to be a proper subset of set ; and inlusion of in is alled to be a strit inlusion. The standard notation (DIN 5473), whih is widel used, refers to the ase of a strit inlusion: that means that and. For improper inlusion, the notations and also are used. union The union of two sets and is a set whih onsists of all elements of set and set. The onlusions whih an be made about the elements of these sets: if a, then a if b, then b if, then and/or Smboliall, the definition of the union of two sets is given b an epression: The operation union is refletive: = = { a a and / or a } intersetion The intersetion of two sets and is a set whih onsists of all elements whih simultaneousl belong to set and set : = and { } The onlusions about the intersetion: if, then and and, onsequentl, if and, then The operation intersetion is also refletive: =
6 Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 differene \ omplement \ The differene of two sets and is a set whih onsists of all elements of set that do not belong to set (reads without ): \ = { and } The onlusion about differene: if \, then and if and, then \ Differene is not a refletive operation, in general: \ \ If set is a subset of set, then the differene \ is alled a omplement of set with respet to set : { } = \ = and Some properties of the omplement of set with respet to set : = ( ) = = = = produt produt (or Cartesian produt) of two sets and is a set onsisting of all a,b where a and b : a ( a,b) ordered pairs ( ) {( ) } = a,b a,b Cartesian produt is not refletive, in general, Cartesian produts of higher orders are reeived b onseutive appliation of b Cartesian produt to more than two sets: and so on. {( ) } C = a,b, a,b, C We will denote the Cartesian produt of set with itself b the usual notation and, in general, = n = n times Properties of set operations: ) = ommutative law = ) ( ) C = ( C) assoiative law ( ) C = ( C) 3) ( C) = ( ) ( C) distributive law ( C) = ( ) ( C) 4) \ ( C) = ( \ ) ( \C) De Morgan s rules \ ( C) = ( \ ) ( \C)
Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 7 3. PROOFS: It is important to know how statements in set theor are proved. Here, we provide some eamples of proof of statements in set theor. Eample (diret proof) Prove the ommutative law for intersetion: = Proof: To prove that two sets are equal we need to show the validit of two statements: ) Let a, then, aording to the definition of intersetion, a and a, this also means, that a and a (in mathematial logi, the statement and is equivalent to the statement and ). Therefore, using the definition of intersetion again, we have that a. So, if a, then a. That means that is a subset of, or, in other words, that. ) the proof of this part is ompletel similar to the previous one. From ) and ) and the aiom of etensionalit, it follows that = q.e.d. Eample Proof: (proof b ontradition) Prove that for an set (empt set is a subset of an set) ssume that the statement we have to prove is not true, i.e. that the empt set is not a subset of an set; it means that there eists at least one set suh that is not a subset of ; it means that there eists an element a suh that a ; but the empt set ontains no elements, therefore, what we get ontradits the definition of the empt set; Therefore, our assumption that is not a subset of an set is not true; that means that the opposite statement is true: is a subset of an set. q.e.d.
8 Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 4. NUMERS: What we in the previous setions were abstrat sets with elements smboliall denoted b letters. Here, we onsider eamples of the sets of ommon numbers, though we are not going to give strit definitions of these sets. natural numbers = set of natural numbers, = {,,3,... }, numbers whih appeared in ounting and an be used in indeing ( a,a,a 3,... ). integers = set of integers, = {..., 3,,,0,,,3,... }. rational numbers = set of rational numbers, a = a, b. b real numbers = set of real numbers, set of all numbers whih are represented b the positive or negative infinite deimals suh as 35.967 or.593... Remarkable real numbers: =.44..., e =.788..., π = 3.39.... = {,, } {( ) } The set ( ) represents the Eulidian plane and the set 3 =,,z,,z represents the 3-dimensional Eulidian spae. omple numbers = set of omple numbers, = { a + ib a,b, i = }. Field of Numbers The algebrai struture of the sets of numbers is formulated in the definition of the abstrat algebrai notion of a field. Field is a set F with two binar operations: addition ( + ) and multipliation ( ) for whih the following aioms are satisfied: ) if, F, then + F (losure for addition) ) if, F, then F (losure for multipliation 3) + = + for an, F (ommutative law) = for an, F (ommutative law) 4) + ( + z) = ( + ) + z for an,,z F (assoiative law) 5) ( z) = ( ) z for an,,z F (assoiative law) 6) ( + z) = + z for an,,z F (distributive law) 7) there eists a unique element 0 F suh that + 0 = 0+ = for an F (zero element) 8) there eists a unique element F suh that = = for an F (unit element) 9) for an F there eists a unique number F suh that + = + = 0 (negative of ) 0) for an F ( 0) there eists a unique number F suh that = = (inverse of ) The set of rational numbers, the set of real numbers and the set of omple numbers are fields. ll of them posess the abovementioned properties.
Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 9 For the negative of, we usuall use the notation, and for the inverse of, we use the notation. The negative and the inverse allow us to introdue two new operations with the elements of the field: subtration and division. Previousl introdued sets were just arbitrar olletions of abstrat objets (although we onsidered onl the objets of a mathematial nature). Now, the fields just introdued are sets in whih mathematial operations are applied, and the also satisf the determined onditions. The most signifiant is that fields are omplete sets of the objets with the given properties the inlude all elements neessar for performing the introdued operations. lgebrai Rules Reall the main algebrai rules for real numbers. Let,,z,u,v, then a) + z = + z = z = z, z 0 = b) 0 = 0 = 0 ) = ( ) d) ( ) ( ) = e) = 0 = 0 or = 0 f) g) h) u v+ u + = v v u u = v v v v = = u u v v Powers Let, and m,n, then a) b) ) d) = n times n = n 0 =, 0 n m n m n = + e) m n = m n m f) ( ) n = mn g) ( ) n n n = h) n = n n
0 Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 5. CONSTNTS ND VRILES Two words onstant and variable are used for representation of the elements of a set of numbers: onstant = partiular element of a set; variable = a smbol representing the elements from some set, whih is alled a range of a variable. For eample, some fied element a from a set is a onstant. The smbol whih an assume designation of all elements of the set is a variable. We alread used variables to illustrate the definitions of operations with sets. The most tpial usage of a variable in set theor looks like: for an whih means for an element in set. smbol = The smbol = (equal) will be used in two different senses: identit a = b means that onstants a and b represent the same element; equation = b means that there eists an element b whih an be used for replaement of variable. In this ase, we an sa that variable assumes the value b, or that the solution of this equation is b. Properties of the equal sign: ) = refleivit ) if =, then = smmetr 3) if = and = z, then = z transitivit Equal is a partiular ase of a more general equivalene relation.
Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 6. COORDINTES Cartesian Coordinates l We alread adopted the notation for the set of all real numbers. The geometrial figure assoiated with is a line. We fi a point 0 (whih we all the origin) in the line l, then we hoose the unit of length as a distane between origin and some other point (whih we denote ) to the right of the origin. The real number 0 orresponds to the origin, and the real number orresponds to the point alled. Then an other point a to the right of the origin orresponds to a positive real number whih quantitativel is equal to the length of the segment between 0 and a in terms of the hosen unit length. The points to the left of the origin orrespond to negative real numbers in the same manner. s a result, we have that for an real number a, there eists a unique point on the line; and for an point on the line there orresponds a unique real number. Therefore, we an establish a one-to-one funtion from l into. This funtion whih we denote b :l are Cartesian oordinates of the points on the line eah point on the line has a orresponding numerial value. The line l we all the real line, the diretion to the right of the origin is a positive diretion of the line whih is denoted b an arrow. In terms of Cartesian oordinates, the line is said to be an -ais. The real numbers on the -ais are ordered: if a and b are two distint points on the real line, and point a is to the left of the point b, then a<b. Using this fat, we an introdue the following sets (intervals): ( a,b) = { a < < b} open interval; [ a,b] = { a b} losed interval; (,a) = { < a} open; (,a] = { a} half-open; ( b, ) = { > b} open; [ b, ) = { b} half-open. Here, we introdue the smbols and - whih do not orrespond to an point on the real line or to an real number. Now, onsider the Cartesian produt ( ) The geometrial figure assoiated with the set { a,b a,b } = =. is a plane. n point in the plane an be put in a one-to-one orrespondene with the set b introduing Cartesian oordinates. The figure shows an orthogonal Cartesian oordinate sstem. Lines and are alled the -ais and -ais, orrespondingl. The value is alled an absissa, and the value is alled an a,b, the absissa value is alwas given first. We ordinate. In the oordinates ( ) will use the orientation of the ais alwas as shown in the Figure (right-handed sstem).
Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 3 3 The set of all triplets of real numbers ( ) { a,b, a,b, } = is assoiated with 3-dimensional spae. This assoiation an be established b the orthogonal Cartesian oordinate sstem with three aes: absissa, ordinate, and appliate z, whih for a right-handed sstem has a form shown in the Figure. The idea of a Cartesian oordinate sstem seems to be ver simple, but the disover of this sstem took a long time. Though the origin of mathematial siene reahes deep into the enturies.c., the introdution of Cartesian oordinates was made b the Frenh mathematiian and philosopher Rene Desartes (Latin name Cartesian) onl in the 7 th entur. It was, probabl, the greatest revolution in mathematis sine the ahievements of Greek mathematiians. We use Cartesian oordinates to desribe the position of a point in spae. There are also other oordinate sstems whih an be used for the same purpose. In some ases, the are more onvenient and simple than Cartesian oordinates. Here, we list these sstems and show their relation to the Cartesian oordinate sstem. polar oordinates in (plane polar-oordinate sstem) For polar oordinates in plane, we need a referene diretion. It is onvenient for further onversion, to hoose the positive diretion of the -ais 0 as the referene diretion. Then an arbitrar point in the plane P is assoiated with the r,θ where r is the distane between point P and the origin 0, and ordered pair ( ) θ is a prinipal angle measured in radians in a ounterlokwise diretion between the referene diretion 0 and the diretion of 0P. Polar oordinates of point P: ( r,θ ), where r is the radius; r 0 (r<0 an also be used) θ is the polar angle; 0 θ < π onversion polar to Cartesian: Cartesian to polar: = r osθ = r sinθ r = + tanθ = artan θ, if > 0 π, if = 0 and > 0 θ = 3 π, if = 0 and < 0 π + artan θ, if < 0
Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 3 lindrial oordinates in P P 3 (a oordinate sstem whih is of importane for problems with rotational smmetr about one of the ais) Clindrial oordinates desribe the position of the point P in spae b the triple r, θ,z, where of numbers ( ) r θ z is the radius of linder (distane between the point and the z-ais), r 0 is the azimuth, angle between the positive diretion of the -ais and the diretion of 0P, where the point P is the orthogonal projetion of the point P onto the -plane; 0 θ < π is the Cartesian z oordinate of the point P lindrial to Cartesian Cartesian to lindrial = r osθ = r sinθ z r = + tanθ = = z z = z spherial oordinates in P P 3 (a oordinate sstem whih is of importane for problems with rotational smmetr about the origin) Spherial oordinates desribe the position of the point P in spae b the triple r, φθ,, where of numbers ( ) r is the radius, a distane between the point P and the origin 0 ; r 0 θ φ is the latitude, the angle between the diretion of 0P and the positive diretion of the z-ais; 0 θ π is the longitude, the angle between the positive diretion the of -ais and the diretion of 0P, where the point P is the orthogonal projetion of the point P onto the -plane; 0 φ < π spherial to Cartesian = r sinθosφ = r sinθ sinφ z = r osθ Cartesian to spherial r = + + z z z osθ = = r + + z tanφ = spherial to lindrial ( ρθ,,z) lindrial to spherial ( r, φθ, ) ρ = r sinθ ρ = r + z θ = φ φ = θ z ρ = r osθ θ = artan z
4 Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 7. FUNCTIONS Definition Let and be two nonempt sets (whih an be distint or the same). funtion f of set into set is a mapping whih for ever element of set gives in orrespondene preisel one element in set. domain = f ( ) range o domain Notation for a funtion: f : = f ( ),, Here, is an independent variable, and is a dependent variable the value of the funtion f at. We also all an image of under the mapping f. Set is said to be a domain of funtion f, and the set is the o-domain. The set of all images R is alled the range of funtion f: R = { = f ( ), } { (,f ( ) ), f ( ) } This definition of a funtion is traditional in alulus. The other (more formal) definition: Let and be two nonempt sets. Funtion f : f =, is a set of ordered pairs {( )} suh that: ) for an, there eists suh that (,) f ; ) if (,) f and ( u,v) f, and if = u, then = v. This definition also provides a basis for the graph of the = f whih is the set of points with the funtion ( ) oordinates ( ( )) {, f }. = The real valued funtion = f ( ) of the real variable is usuall defined b the equation without indiation of the domain, assuming that the domain is the = f is defined. For eample, largest subset of for whih the funtion ( ) if the funtion is given b the equation = then its domain is = (,) and its range is R [, ) funtion is shown on the left. =. The graph of this onstant funtion funtion whih for all assumes the same value a. onstant funtion is given b the equation: = a inverse funtion Let f be a funtion from into, and let Y be a range of the funtion f. If = f ( ) = f ( ) range Y there eists a funtion g from Y into, suh that ( ( )) g f = for all, then the funtion f is said to be invertible and the funtion g is alled the inverse of the funtion f. Notation is used for the inverse funtion: ( ) = f f
Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 5 one-to-one funtion The funtion f : is alled one-to-one if from follows f ( ) f ( ) for all,. f f ( ) Reall the horizontal line test for a one-to-one funtion. f ( ) Theorem The funtion is invertible if and onl if it is one-to-one. monotoni funtions Proof: ) (if a funtion is invertible, then it is one-to-one) Let, be suh that. Suppose that f ( ) = f ( ), then f ( f ( ) ) = f ( f ( ) ) given that the inverse funtion eists. nd, onsequentl, from the definition of the inverse funtion, we have that = that ontradits the ondition that, therefore, our assumption that f f f f ( ) = ( ) is wrong, and we have that ( ) ( ) when, that means that f is one-to-one. ) (if the funtion is one-to-one, then it is invertible) The monotoni funtions are the funtions of one of the following tpes: a) f ( ) is an inreasing funtion on the interval ( a,b ) if f ( ) < f ( ) for all, ( a,b) < b) f ( ) is a non-dereasing funtion on the interval ( ) f ( ) f ( ) for all, ( a,b) < ) f ( ) is a dereasing funtion on the interval ( a,b ) if f ( ) > f ( ) for all, ( a,b) < d) f ( ) is a non-inreasing funtion on the interval ( ) f ( ) f ( ) for all, ( a,b) < suh that a,b if suh that suh that a,b if suh that Theorem If the funtion f ( ) is either inreasing or dereasing on ( ) then f ( ) is invertible on ( a,b ). omposition Let R be the range of the funtion f ( ), and let the funtion g( ), R R is the funtion a,b, be defined on R. Then the omposition of f and g ( ) g f = g f f g Eample 3 Let f ( ) = and ( ) Then ( ) g f = g f = + g = + ( ) g f = g f Note, that the omposition ( ) ( ) f g = f g = + Therefore, in general, f g g f.
6 Chapter I MTH FUNDMENTLS I. Sets, Numbers, Coordinates, Funtions ugust 30, 08 8. REVIEW QUESTIONS: ) What are sets? ) What are two basi was to desribe the elements of sets? 3) Reall the operations with sets. 4) Reall the sets of numbers? 5) Whih sets of numbers are the subsets of other sets of numbers? 6) Wh the sets of natural numbers and the set of integers are not the fields? 7) What is the differene between onstants and variables? 8) What is the differene between the identities and equations? 9) Reall the definition of major oordinate sstems. 0) Reall the definition of funtion. ) What monotoni funtions are one-to-one? ) If funtion satisfies the horizontal line test, is it invertable? EXERCISES: ) Prove the following properties of the operations with sets: a) = ommutative law for union b) = ommutative law for intersetion ) Prove or disapprove the following statement: If =, then = 3) Prove that if from f ( ) = f ( ) follows that =, then funtion f is invertible. 4) Prove or disapprove the following statement: Let,. If + + is even, then is odd. 5) Prove that if a funtion is one-to-one, then it is invertible.