1、Sets and FunctionsDRSets 集合集合 and Functions 函数函数2In this section,we will introduce the concepts of set and function,which are the very beginning of Advanced Mathematics.The Concepts of Sets3The relation means that is an element of set A,read as“belongs to ”.aA aaAThe relation or means that does not
2、belong to .aA aA aA A set 集合集合 is a collection of all objects which are sharing some properties.Each of the objects belonging to a set is called an element 元素元素 of the set.(Notations)Sets are always denoted as capital letters,such as A,B,.The elements of a set usually are denoted as small letters,su
3、ch as a,b,.Finite Set and Infinite Set4 A set consisting of finite number of elements is called a finite set 有限集有限集.A set consisting of infinite number of elements is called an infinite set 无限集无限集.A set consisting of all elements under consideration in a given discussion is called universal set 全集全集
4、,denoted by A set containing no element is called a empty set 空集空集,denoted by.Finite Set and Infinite Set5:the natural number set自然数集自然数集 is denoted by N.:the positive integer set 正整数集正整数集 is denoted by N+.:the rational number 有理数集有理数集 is denoted by Q.:the real number 实数集实数集 is denoted by R.:the int
5、eger set 整数集整数集 is denoted by Z.is a natural number|0,1,2,Nx x is a positive integer|1,2,Nx x is an integer|0,1,2,Zx x is a rational number|Qx x is a real number|Rx x Comparing Sets6 Let A and B are two sets.If each element of A is element of B,then A is called a subset 子集子集 of B,denoted byBA “conta
6、ins ”.BA AB or ,BAread as“is contained by ”orXB AAB Comparing Sets7 If and ,then A and B are called equal 相等相等,denoted by .AB AB AB If and ,then A is called a proper subset 真子集真子集 of B,denoted by or .AB AB AB AB Set21,2,|320ABx xxABOperations on Sets8Basic operations on setsLet and are two sets.AB T
7、he union并并 of A and B is a set,which contains all elements of A and B,is denoted by ,this means that|or.ABx xAxB AB ABOperations on Sets9 The of two set A and B is a set,whose elements are those belong to both A and B,is denoted by ,this means thatABI|and.ABx xAxBIRemark If ,then and are said to be
8、disjoint.AB IABABOperations on Sets10 The difference 差差 of two set A and B is a set whose elements are those belongs to both A but not to B.Denoted by ,this means thatA B|and.A Bx xAxBABOperations on Sets11 If ,then the difference is called the complement 补补 of B with respect to A,denoted by .BA A B
9、AC BRemark If X is the universal set,is called the complement of ,denoted by or XBBBC.cBBAC BRules of Operations on Sets12Let ,be any three sets,thenAB C(1)Commutative law;ABBAABBAUUII(2)Associative law(3)Distributive law ;ABCABCABCABCUUUUIIII ;ABCACBCABCACBCA BCACBC UIIUIIUUIUIIIRules of operations
10、 on sets13(4)Idempotent law;AAAAAAUI(5)Absorption law;AAA UIIf thenAB;.ABBABAUI(Dualization law)If A and B are two sets of the universal set X,then();().ccccccABABABABUIIUCartesian Product of Sets14 The Cartesian Product 笛卡尔积笛卡尔积 (product,or direct product,or cross product)of two sets A and B is a s
11、et with the element as(x,y),where x and y are elements of A and B respectively.The product of two sets is denoted as means thatAB(,)|,.ABx yxA yB Let ,then|Ax axb|By cyd(,)|,.ABx yaxb cydProperties of Real Number Set15 Real Number SetProperty 1 (Closure)Under rational operations,the number obtained
12、by performing some rational operations on any two real number is also a real number.Let ,then ,a bR,/.ab a b a bRProperty 2 (Order)Any two real number ,one and only one of the following relations hold:,a bR,.abababProperties of Real Number Set16Property 3 (Density)There must exists another real numb
13、er between any two real number.Property 4 (Completeness)If we put all the real number on the coordinate axis,then they will fill the axis.The rational number can not fill the axis but real number can.xMappings and Functions17:,or:(),fABfxyf xxAa aLet A and B be two non-empty sets.If for every ,there
14、 exist a unique corresponding to x xAyBaccording to some determined rule f,then f is called a mapping of A into B denoted byHere,y is called the image 像像 of x under the mapping f and x is the inverse image原像原像 of y.Set A is called domain of definition 定义域定义域 of mapping f,denoted by D(f).The set cons
15、isting of the image y of all element is called thexArange 值域值域 of f,denotes by R(f)or f(A).Mappings and Functions18Remark A mapping is also called an operator 算子算子.If ,then the mapping is also called a function 函数函数.RB BAf:If ,then f is called a transformation 变换变换 on the set A.BA If a mapping f map
16、s every element of A into itself,then fis called the identity mapping 恒等映射恒等映射 or unit mapping 单位映射单位映射,denoted by IIAor,that isxIxAx,Mappings and Functions19(Function 函数函数):(),.fxyf xxAa aLet A and B be two non-emptysubsets of the real number set R.Then a mapping:fABis called a function of a single
17、 variable,denoted by where x is called independent variable 自变量自变量 and y is called dependent variable 因变量因变量.f(x0)is called function value 函函数值数值 at x=x0.A is domain of definition 定义域定义域 of f,denoted by D(f)and),(|)(AxxfyyAf is called range 值域值域 of f,denotedby R(f).Mappings and Functions20D=domain s
18、etR=range setA function from set D to set R is a rule that assigns a singleelement of R to each element in D.Input(Domain)xOutput(Range)f(x)f:each input in the functions domain has only one output in the range.Notes on Functions21Domain of definition Like the mapping,domain of definition of function
19、 is one of the essential factors in the concepts of function.To find the domain of definition of a function is to find where the rule f can be hold.If the domain of definition is an interval a,b,then it is often called the interval of definition 定义区间定义区间.Notes on Functions22 It is well known that th
20、e interval a,b means the set of all real number between a and b,including a and b.That is,|a bx axboxab(,)|a bx axboxabWhile(a,b)means the set of all real number between a and b,not including a and b.Notes on Function23Remark A f u n c t i o n d e p e n d s o n l y o n t h e and the.Function is inde
21、pendent of the symbols of its variable.Equal Functions 2422(1)()2;()2.(2)()ln,()ln,()|ln|.f xxg ttf xx g xx h xx If functions f and g have the same domain of definition,and for any x D(f)=D(g),we have f(x)=g(x),then f and g are said to be equal 相等的相等的.Judge whether the following pair of functions ar
22、e equal:Solution(1)Yes (2)No.Examples25 Let101(),212xf xx find the domain of definition(3).f x Solution1031(3)2132xf xx 101()212xf xx Q132221xx So()3,1D f Finish.of functionExamples26()()(1)1f xf x fx(1)(1)(1)11fff Let function:,f RRand suppose()()()f xyf x f yxyfor every,.x yR Find the representati
23、on of f(x).Solution Set y=1,we haveTaking x=1,we obtainor2(1)(1)2.ffSo,(1)2(1)1.forf Thus1()1()(1).2f xxorf xx Finish.1()(1)2f xx is not the solution since the value of the required function at x=0 is 1 or 0.Express a Function27Correspondence RuleTabulation Tabulate all the values of independent var
24、iable x and dependent variable y.Graph Draw a graph of x and y to show the relation.Analytic representation Use trigonometric functions,inverse trigonometric functions,exponential functions,logarithm functions and power functions to express the relation.Basic functions28Constant functionsy=Const;Tri
25、gonometric functionsy=sin x,y=cos x,y=tan x,y=cot x,Inverse trigonometric functionsy=arcsin x,y=arccos x,y=arctan x,y=arccot x;Exponential functionsy=ax,(a 0,a 1);Logarithm functionsy=logax,(a 0,a 1);Power functionsy=xa.Some functions and their graphs29 Some basic functions and their graph1)Constant
26、 function:()2,()(,)yf xD f 2)Piecewise defined function:,0()|,0 xxyf xxxx -1-0.50.510.20.40.60.81Some functions and their graphs30 1 2 3 4 5 -2-4-4-3-2-1 4 3 2 1 -1-3xyothis function gives the largest integer number smaller than or equal to the given real number,denoted by .x3)The greatest integer f
27、unction or floor function:Some functions and their graphs314)The sign function:1,0sgn0,01,0 xyxxx yx1-1Some functions and their graphs32 ,then the integer variable function can be denoted as a sequence:5)Dirichlet function:1,()0,xQyD xxR Q 6)Integer variable function:If we write()nf na,nN 12,na aaLL
28、.A function is defined on the set of positive integer and denoted by(),yf nnN.Some functions and their graphs337)Maximum function:max(),()yf xg x.yxo)(xf)(xgSome functions and their graphs348)Minimum function:min(),()yf xg x.yxo)(xf)(xgPrimary Properties of Functions35BoundedLet()yf x and()D f is th
29、e domain of definition of function f.Let()AD f.If for any ,xA 0M,so that(s.t.)always 0 xA,s.t.then we call function f is unbounded 无界的无界的 on A.|()|f xM,then we call function f is bounded 有界的有界的 on A.Otherwise,if for any 0,M|()|,f xM Primary Properties of Functions36M-Myxoy=f(x)ABoundedUnboundedM-Myx
30、oA0 xPrimary Properties of Functions37MonotoneLet().AD f If and ,have 12,x xA12xx 12()()f xf x (or 12()()f xf x ),Let()yf x and()D f is the domain of definition of function f.we call function f is monotone increasing 单调增加的单调增加的(or monotone decreasing 单调减少的单调减少的)on A.12()()f xf x (or 12()()f xf x),If
31、 we call function f is strictly monotone increasing 严格单严格单调增的调增的(or strictly monotone decreasing 严格单调减的严格单调减的)on A.Primary Properties of Functions38()yf x)(1xf)(2xfxyoIStrictly monotone increasing()yf x)(1xfxyoI)(2xfStrictly monotone decreasingPrimary Properties of Functions39Even and OddLet()yf x a
32、nd()D f is the domain of definition of function f.If()xD f ,we havethen,we call function f is an even function 偶函数偶函数.()()fxf x,then,we call function f is an odd function 奇函数奇函数.If()xD f ,we have()()fxf x ,Primary Properties of Functions40yx)(xf )(xfy ox-x)(xfEven function)(xf yx)(xfox-x)(xfy Odd fu
33、nctiony1x1y=x+1NeitherPrimary Properties of Functions41Periodicity()yf x Let and()(,)D f is the domain of definition.of function fIf 0T,so that()xD f ,we have()(),f xTf xthen,we call function f is a periodic function 周期函数周期函数.2T 2T32T 32TPrimary Properties of Functions422(),()ln(1),xxf xaag xxx Supp
34、ose prove that f is an even function and g is an odd function.Proof2()ln(1)g xxxQ Q()xxf xaa Q Q()xxfxaa 2()ln()1)gxxx()(),f xfxthen f is an even function.21ln()1xx 2ln(1)xx ()(),gxg x then g is an odd function.Finish.Inverse functions 反函数反函数43AB,xA 1yB ().yf x,yB 1xA (),f xy (One to One Mapping 一一映
35、射一一映射)Suppose that f :AB is a mapping.corresponds to x,and If corresponds to y as well and satisfiesthen f is called a one to one mapping from A to B,and the sets A and B are said to be in one to one correspondence.Inverse functions44Suppose that:fAB is a one to one mapping,yB ,1xA such that()f xy.T
36、hen we can define a mapping:g BA which maps each yB to xA,that is()g yx.The mapping g is called the inverse mapping of the mapping f,denoted by 1gf ,and f is said to be invertible 可逆的可逆的 mapping.(Inverse Mapping逆映射逆映射)Inverse functions45Let the function:()fAR f,where A and()R f are both real number
37、sets,regarding f as a mapping.If the inverse mapping 1:()fR fA exists,then 1f is called the inverse function of f.The operation of finding 1f from f is called the inverse operation.(Inverse Function 反函数反函数)Inverse functions46For example,if 21()(1)1xyf xxx then11()(2)2yxfyyy Interchanging x and y,we
38、have11()2xfxx RemarkObserve that x=f-1(y)is what we obtain by solvingwith the reciprocal function 1.()f ythe equation y=f(x)for x in terms y.It must not be confused Inverse functions47Theorem(Existence theorem for inverse function)If funtion f is a strictly monotone increasing(or decreasing)function
39、 on the set A,then there must exist an inverse function 1f,and the function 1f is also a strictly monotone increasing(or decreasing)function on the corresponding range()R f.Inverse functions48()yf x xyo(,)Q b a(,)P a b1()yfx yx Inverse functions492,(,1),(),1,4,2,(4,).xx xyf xxxx Find the inverse fun
40、ction of the piecewise functionSolutionSince f(x)is a strictly monotone increasing function on(,),we have 1f exists.1)while,(,1),yx x ,(,1);xy y we have2)while 2,1,4,yxx,1,16;xy ywe have3)while 2,(4,),xyx2log,(16,).xy ywe haveInverse functions50So,2,(,1),1,16,log,(16,).y yxy yy y Solution(continued)
41、Therefore,the inverse function of f is2,(,1),1,16,log,(16,).x xyx xx x Finish.2,(,1),(),1,4,2,(4,).xx xyf xxxx Find the inverse function of the piecewise functiondetermined by the mapping g and f.Hence,a new mapping from:fg ACo.,there hasLet:g AB:fBC.Then for each xA,which corresponding to y.an uniq
42、ue()()zf yf g xC,()yg xBan unique which corresponding to x.Again,there haszC xA,there exists an unique Thus,for each can beA to C is determined.This new mapping is called a compositemapping of g and f,denoted by Composition Functions 复合函数复合函数51 (Composite Mapping 复合映射复合映射)Composition Functions52ABCf
43、f g gAlso the composite mapping can be written as:()()(),.fgxf g xxA That isComposition Functions53 (Composite Function 复合函数复合函数)Let the domains of definition of the function f and g be D(f)=B xA,a value u can be determined by the function g denoted bycomposite function of function g and f.,then for
44、 any()R gB and D(g)=A respectively.If the range of g,uB,a value y can be determined by theu=g(x),and since a value y can bexA function f,denoted y=f(u).Hence for anyxyadetermined by function g and f.This new function is calledAlso the composition function can be wrote as()(),yfgxf g xxA.Composition
45、Functions54fgxg(x)f g(x)()D gA()R g()D fB CgfTwo functions can be composed when the range of the first lies in the domain of second.Composition Functions55Remark If()R g is not contained in()D fB,then the function g can not be composed with the function f.However,if the domain()D g is restricted to
46、a set()AD g%such that for anyxA%,()()g xuD f,then the function g can be composed with f:()()yfgxf g x.A%()R g()D fABCggffThe domain of definition of this composite function is the set.A%Composition Functions56()2,),R u Not any two functions have compositions.arcsinyu,22uxthere does not have composit
47、ion.Since D(y)=-1,1,and2arcsin(2)yxFinish.we have,LetComposition Functions57For instance,:sinfxxa,:g xxa,can be composed into :sin,0,)fg xxxoaRemark Whether f and g can be composed into f g depends only on the relation R(g)and D(f),and does not depend on the symbols for the variables of these functi
48、ons.Composition Functions58:sin,:cos,:lnfxxg xxh xxaaa()(,)()(,)R hD g ()1.1()(,),R gD f ()()sin(cos(ln),()(0,)fg h xf g h xxxD hSince,andwe can form the composite function.Remark Composition of functions can be extended to any finite number of functions,such asComposition Functions59Solution2()()fx
49、f f x 2()1()f xfx 222111xxxx 212xx 222()1()fxfx 22212112xxxx 213xx By mathematical induction,we can prove thatFinish.If 2()1xf xx ,show.()()nnfxf ff x 14444444 42 44444444 3LL2()1nxfxnx .32()()fxf fx Elementary Functions and Hyperbolic Functions 双曲函数双曲函数6021yx3sincosyxxcot3xy A function formed by th
50、e six kinds of basic functions by a finite number of rational operations and compositions of functions which can be expressed by a single analytic expression is called elementary function.(Elementary Functions 初等函数初等函数):Hyperbolic Functions61Hyperbolic sinesinh,(,)2xxeexx Hyperbolic Functions62Hyper
