1 Sets of real numbers

Transcription

1 1 Sets of real numbers Outline Sets of numbers, operations, functions Sets of natural, integer, rational and real numbers Operations with real numbers and their properties Representations of real numbers Elements of topology We denote by N = {0 1 } Z = { } Q = : Z, 6= 0ª (the R (the set of natural numbers) (the set of integer numbers) set of rational numbers) (the set of real numbers) In the following chapters, we will also be concerned with the sets R, R, or more generally R ( 1), defined as follows R = { =( 1 ): 1 R} Given a point =( 1 ) R, 1 are called the coordinates of. In the cases =1 and we can represent by a point on the real line, in the real plane, respectively in the real space, as indicated in the figure below. In the cases 4 we cannot represent graphically the point (it is difficult for example to make a sketch in a 4-dimensional space!), but we can still think abeingapointinr. R R R x x x x 0 x 0 x 1 x 0 x 1 Figure 1: Graphic representation of a point (vector) in R, R and R. Alternately, the point R can also be viewed as a vector in R, from the origin to the corresponding point (see Figure 1 above). Recalling the addition of vectors, we can define the addition of points in R as follows: if =( 1 )and =( 1 ), we define + by + =( ) (1) There are two multiplication operations which can be performed with a point/vector in R : i) multiplication by a scalar: if =( 1 ) R and R, wedefine the scalar product R by =( 1 ) () ii) dot product: if =( 1 ), =( 1 ) R,wedefine the dot product R by = () 1

2 Using Pythagora s theorem, it is not difficult to see that the length of the vector =( 1 ) R (denoted by ) isgivenby q = = (4) The length k k of a vector in R is called in mathematics the norm of. It has the following properties: Proposition 1.1 For any R we have the following: i) k k 0 ii) k k =0implies =0=(0 0). iii) k k k k k k k k + k k iv) k k = k k v) k k = k k for any 0. Proof. Exercise. y x y x 0 Figure : The distance between the points and is the length k k of the vector. From Figure we can see that the distance between the points =( 1 ), =( 1 ) R is equal to the length of the vector, hence we define q ( ) = = ( 1 1 ) + +( ) (5) Remark 1. Note that in the case =1, the above formula becomes q ( ) = = ( 1 1 ) = 1 1 in which we recognize the usual formula for the distance between two points on the real line: the distance is just the absolute value of their difference. Also, in the cases =and =the above formula becomes the familiar formulae from analytic geometry q ( ) = = ( 1 1 ) +( ) respectively ( ) = = q ( 1 1 ) +( ) +( ) Thus,ingeneral,wewillwritek k for the distance between two points in R (perhaps in the case =1it is more common to write instead k k, although both notations are acceptable). Often, we are interested in describing the points which are close to a given point R. Recalling that the distance between and is just k k, this leads to the following definition:

3 R R R x r B(x, R) x B(x, R) B(x, r) r 0 x x r x x + r 0 1 Figure : The ball ( ) centered at of radius 0inR, R and R. Definition 1. We define the open ball with center R and radius 0 by ( ) ={ R : k k } (6) Remark 1.4 In the case =1,weobtain ( ) ={ R : } =( + ) so in this case the ball is just the interval centered at of radius. In the case =,weobtain n ( ) ={ R : } = =( 1 ) R :( 1 1 ) +( ) o so in this case the ball is just the disk centered at of radius. In the case =,weobtain n ( ) ={ R : } = =( 1 ) R :( 1 1 ) +( ) +( ) o so in this case the ball is the usual -dimensional ball centered at of radius. In the higher dimensional cases ( ), even though we cannot represent graphically the ball ( ), wewill still refer to ( ) as the open ball of center and radius. Thenotionofanopen set, extends the notion of an open interval. Recall that in R, anopeninterval( ) is the set of all points between (and not including) the endpoints and, thatis ( ) ={ R : } What makes the interval ( ) open, is the following property: whichcanalsobewritteninthefollowingform ( ) 0 ( + ) ( ) ( ) 0 ( ) ( ) Inthisform,thisisexactlythedefinition of an open set in R : Definition 1.5 Aset R is called open if for any point there exists an 0 such that ( ). Aset R is closed if the complement = R is an open set. A neighborhood of a point R is any set = R for which there exists 0 such that ( ). Note that by definition, the empty set and R are both open sets and closed sets. Example 1.6 For any R,, the open interval ( ) is an open set, and the closed interval [ ] is a closed set. Note that the intervals ( ] and [ ) are neither open nor closed. Conversely, it can be shown that a set R is open if and only if it is a (countable) union of open intervals, that is = 1 ( ).

4 Some properties of the open/closed sets are contained in the following: Proposition 1.7 Finite intersections of open sets and arbitrary unions of open sets are open. Finite unions and arbitrary intersections of closed sets are closed sets. Proof. Exercise. The following definition is meant to indicate the relative position of a point with respect to a set. Definition 1.8 Apoint R is called an: i) interior point of if there exists 0 such that ( ) ; ii) closure point of if for any 0 ( ) 6= ; iii) boundary point of if for any 0, ( ) 6= and ( ) * ; iv) accumulation point / limit point of the set if for any 0, ( ( ) { }) 6= ; v) isolated point if but is not an accumulation point of. Remark 1.9 The notion of accumulation / limit point of a set has the following equivalent definition. A point R is a limit point of if and only if there exists a sequence 1 { } of points in { } such that lim =. In other words, a limit point point of is a point which can be obtained as the limit of points in { } (both requirements that and 6= are essential here). Definition 1.10 Given a set R,wedefine: i) the interior of the set (denoted ) as the set of all interior points of, thatis = { R : interior point of } ii) the closure of the set (denoted ) as the the set of all closure points of, that is = { R : closure point of } iii) the boundary of (denoted ) by: = Remark 1.11 It can be shown that is the largest open set contained in, moreprecisely = and is the smallest closed set containing, that is We have the following: = [ \ Proposition 1.1 Givenanarbitraryset R, the interior is an open set and the closure isaclosedset, and we have: Example 1.1 For any R with, the interior of the set ( ] is ( ) and its closure is [ ]. For the set =(1 ] {}, thesetofinteriorpointsis =(1 ), the set of closure points is =[1 ] {}, the boundary points are =1, and, the set of accumulation points is [1 ], and the only isolated point is =. 4

5 1.1 Functions An important notion in mathematics is that of a function. Recall that a function is a rule which assigns to each number in a certain set (called the domain of the function) a unique number in a certain set (called the range of the function). If denotes the rule which describes the correspondence between the points in the domain and the range of the function, it is customary to write : (read as defined on with values in ) for the corresponding function. In this notation, ( ) representsthepointin which corresponds to the point. Note that according to the definition, a function is a triple ( ), that is a rule, a domain, and a range. In particular, two functions 1 : 1 1 and : are equal if and only they have the same domain, the same range, and the same rule, that is 1 =, 1 = and 1 ( ) = ( ) for all 1 =. The graph of the function : is the set = {( ( )) : }. In the case when R (or R ) and R, we can represent graphically the graph, by plotting the points with coordinates ( ( )) for all. Given two functions : and : (note that the range of equals the domain of ), we may define the composition of and as the function :, ( )( ) = ( ( )),. The composition of functions is associative (meaning that ( ) = ( ), whenever the composition of and makes sense), but is not in general commutative (meaning that 6= ). For a given function : it may happen that distinct points are mapped into distinct points ( ) - when this happens, we say that the function is injective. Also, it may happen that any point in the range of the function is the image under of a point in its domain - when this happens, we say that is a surjective function. The formal definitions are the following. Definition 1.14 We say that the function : is i) injective if or equivalently ii) surjective if 1, 1 6= = ( 1 ) 6= ( ) 1, ( 1 )= ( )= 1 = s.t. ( ) = iii) bijective if it is both injective and surjective. The important consequence of the fact that a function : is bijective is that to each corresponds exactly one such that ( ) = (why?). This allows to define the inverse function of, denoted by 1 :, bydefining 1 ( ) = if ( ) =. If is a bijective function, then 1 ( ) = for all (7) and 1 ( ) = for all (8) Next, we present a brief overview of the most common mathematical functions Linear functions A linear function is a function : R R, ( ) = +, where R are constants. The graph of a linear function is a line, and it can therefore be sketched by ploting two arbitrary distinct points ( 1 ( 1 )) and ( ( )), and joining them by a line. Note that in the case when the domain of the linear function is just a subset of R (instead of all real numbers), the graph of the function consistsonlyofapartofaline(insteadofawholeline). Recall that the slope of a line is defined as the tangent of the angle between the line and the horizontal axis, that is =tan. (9) If ( 1 1 )and( ) are two points of the line, simple geometric consideration show that the slope can be found by using the formula = 1 (10) 1 5

6 If two lines 1 and have slopes 1, respectively and they are parallel, then 1 =. If they are perpendicular, their slopes verify 1 = 1. Finally, note that when the equation of the line is written in the form = +, its slope is exactly the coefficient of, thatis =. Example 1.15 Sketch the graph of the function : R R, ( ) = +. Redo the exercise in the cases when :[0 ) R and :( 1 ] R. Conversely, if the graph of a linear function is known, we can find its equation by using the formula for the equation of a line determined by two distinct points ( 1 1 )and( ) 1 = 1 (11) 1 1 or the equation of a line determined by a point ( 1 1 ) and its slope 1 = ( 1 ) (1) Solving the above equations for, the corresponding linear function is given by ( ) =, for in the domain of the function(note that the domain of the function can be read from its graph, but its range cannot be explictly determined from the graph - why? When the range of a function is not explicitly stated, we usually consider the range to be R, R,aso). Example 1.16 Find the equation of the line passing through the points (1 5) and ( 7) Quadratic functions A quadratic function is a function of the form : R R, ( ) = + +, where R and 6= 0(if = 0, the function is linear). Its graph is a parabola: oriented upwards if 0 and downwards if 0, symmetric with respect to the line = with vertex at the point ³ 4 4 ;, -intercepts (points where the graph crosses the -axis): if = 4 0and 1 are the solutions of the equation ( ) =0 1, the parabola crosses the axis at the points with coordinates ( 1 0) and ( 0). Example 1.17 Sketch the function : R R, ( ) = +. Is this function injective, surjective or bijective? Can you modify the domain/range of such that is bijective? What is in this case its inverse function? 1.1. Trigonometric functions For angles 0, the trigonometric functions of the angle are defined as follows. Consider a right triangle with angles =, = and =, asinfigure4below. The trigonometric functions are defined as follows sin = tan = = = The definition is then extended to any angle [0 ) by sin ( ) sin = cos = = cot = if [ ) sin ( ) if [ ) sin ( ) if [ ) = 1 Thesolutionsoftheequation + + =0aregivenby 1 = ± 4 if = 4 0. The same formula also works in the case when 0, but the solutions 1 are in this case complex numbers. 6

7 C π α A α B Figure 4: A right triangle with angle =. and similar formulae for cos, tan and cot (recall the trigonometric circle!). Finally, the definition of sin and cos is extended to any R by periodicity, i.e. sin ( + ) =sin. The tangent and cotangent functions are extended by periodicity (note tan and cot 0 are not defined). Some trigonometric formulae: sin +cos = 1 sin ( ± ) = sin cos ± cos sin cos ( ± ) = coscos sin sin sin ± sin = sin ± cos cos +cos = cos + cos cos = sin + sin = sin cos cos = cos sin sin = 1 cos ( ) cos = 1+cos( ) cos sin The function : [ 1 1], ( ) =sin is bijective. Its inverse 1 :[ 1 1] is called the arcsine (inverse sine) function, and is denoted 1 ( ) =arcsin, inotherwordsfor and [ 1 1] we have sin = arcsin = Similarly, the arccosine (cosine inverse) function arccos : [ 1 1] [0 ]isdefined as the inverse of the cosine function, restricted to the interval [0 ]. 1. Exponential and logarithmic functions The exponential function (with base 0) is the function : R R, ( ) =. When 1 the function is increasing, when = 1 is constant, and it is decreasing when 0 1(seethe graphs below). 7

8 (a >1) y y = a x y = a x y (0 <a<1) x x Figure 5: The graph of = in the case 1 (increasing) and (decreasing). Some properties of the exponential function = + = ( ) = ³ = ( ) = for any 0and R. Fromthegraphsaboveitcanbeseenthatforanyvalueof 0, 6= 1,thefunction : R (0 ), ( ) = is bijective. Its inverse, 1 :(0 ) R, 1 ( ) =log is called the logarithm (with base ) of. Note that this means log = = for any 0, 6= 1, 0and R. Some properties of the logarithms for any 0, 6= 1, 0and R. log ( ) = log +log µ log = log log log ( ) = log log = log log 8

9 1. Exercises 1. Evaluate. h (a) 7 5 i (b) +( ) 4 ( 4) ( 4 5) 0 18 (c) ( 1 ) 5( ) +( ) ( 8) 1 ( ) +( 1 ) 1. Simplify the given expressions. (a) (b) (c) (d) ³ ³ ³ i 1 h (1 ) 1i 1 h ( ) Without using a calculator, determine which of the given number is larger. (a) or 1 6 (b) or 6 49 (c) or (d) 65 or Evaluate the following. (a) (b) (c) : : ( ) + 5 ³ :[( ) 4 7( ) ] [15: 5 +(10 1 : 1 ) 14] ( ) ( ) 14 5 (d) (e) 1 15 : ( ) ( ): 5 9 ( ) 9 71 ³ 7 1 : Factor completely the given expressions. (a) (b) (c) + + (d) (e) 9 ( + ) (f) (g) (h) (i) (j) : p 0 1(6): 8 9

10 6. Simplify the given expressions. (a) (b) (c) (d) (e) Solve the given equations (a) ( 5) = (b) 7 8 = (c) = Solve the given inequalities. (a) ( ) + ( )(+) 1 + ( ) (b) h 1 4 ³ 1 5 (c) ()+( 7)+( 5)+(10)+( 4)+(4) ( 5)+( 7) (4)+(+10)+( ) (d) 7+( ) ( ) 4+(+5) ( 1) (e) 1 4 : 5 6 (f) ( 5):(+5)+(+):(+ 5) ³ ³ ( )( )( 1) :( 5 ) (g) 1 1 (6) (h) [( ) +( ) ] ( 1) 100 +( ) (i) 00(0 005) (99) ( ) (90) i ( +5) 1 9. Consider = {0 1} and = { }. Determine all the functions :, and decide which of these are injective. Are any of these functions surjective or bijective? 10. Consider : R R, ( ) = 1 +and ( ) = +1. Find and. 11. Consider : R R, ( ) = +, ( ) = ( ) =. (a) Find and. Aretheyequal? (b) Find and. Aretheyequal? (c) Find and. Aretheyequal? (d) Find ( ) and ( ). Are they equal? (e) Which of the functions is injective? Surjective? Bijective? (f) For the functions that you determined that are bijective, find the corresponding inverse. 1. Consider the functions : R R defined by ½ +1 1 ( ) = 1 and ( ) = ½ 1 (a) Sketch the graphs of and. (b) Are the functions and injective? Surjective? Bijective? 10

11 (c) Find and. 1. Consider the functions : R R defined by ( ) = 4 +4and ( ) = + +. Show that is not injective and is injective. 14. Sketch the indicated functions (a) : R, =1 where 1 = R, =[1 ), =( 1 ] and 1 ( ) = ( ) = ( ) = +1. Areanytwoofthethreefunctionsequal? (b) : R, =1 where 1 = R, =(0 ), =( 1] and 1 ( ) = ( ) = ( ) =. Areanytwoofthethreefunctionsequal? (c) 1 : { 1 0 1} R, 1 ( ) = and ( ) =. Are these functions equal? 15. Find the equation of the line through the points with coordinates ( 1) and (1 ). 16. Find the equation of the line through the point ( 1) and parallel to the line through (5 ) and (0 4 5). 17. Find the equation of the line through origin and perpendicular to the line with equation = Are the lines + =4and = 4 parallel or perpendicular to each other? 19. Find the point of intersection of the lines + =4and =4 (a) Graphically (b) Numerically 0. Solve the equation + +5=0. 1. Factor the polynomial For which value of do the solutions of the equation = 0 coincide?. Determine the axis of symmetry, the vertex and the -intercepts (if any) of the following quadratic functions, then sketch their graphs. (a) : R R, ( ) = + +1 (b) : R R, ( ) = + +1 (c) : R R, ( ) = + 1 (d) : R R, ( ) = Determine the maximum or minimum of the following quadratic functions. (a) : R R, ( ) = + 1 (b) : R R, ( ) = (c) : R R, ( ) = + + (d) : R R, ( ) = Determine the sign of the following quadratic functions. (a) : R R, ( ) = 6 +5 (b) : R R, ( ) = (c) : R R, ( ) = 5 6 (d) : R R, ( ) = Sketch the graph of the function : R R defined by + ( ) = + 11

12 7. Find the values of and for which the quadratic function : R R, ( ) = + + reaches its maximum value 5 at =. 8. Find the equation of the parabola through the points ( 1 10), (1 ), and ( 5 5). 9. Solve the following inequalities. (a) + +0 ( 1) (b) (c) +6 5 (d) (e) Solve the given systems. (a) (b) (c) (d) (e) (f) ½ 1 1 = = 5 ½ = 4 +5 = 641 ½ = + = 1 1 =5:4 +1 =7:1 ½ 5 1 : 4 +1 : ½ + + =15 5 =6 ½ 10 =48 =0 1. Solve the following systems of equations. ½ + =5 (a) + =7 ½ =0 (b) +6 9 = +. Solve the following systems of inequalities. ½ 7 (a) (b) (c) ½ ( 1) 4( +1) Consider the function :, ( ) = Determine a choice for the sets and such that the function is bijective, and then find its inverse function 1 :. 4. Sketch the graphs of the exponential functions : R R defined by ( ) = and ( ) = Decide which of the numbers 1+ 6 and + is larger. 6. Solve the given inequalities. 1

13 (a) 5 1 ³ (b) µ (c) Solve the following equations. (a) +1 + = 108 (b) +1 +=0 (c) + 1 =8 (d) = 10 (e) = (f) = (g) +1=0 (h) +4 = 7 (i) = 1 8. Solve the following equations. (a) log 5 =1 (b) log +1 + = (c) lg ( +6) = 1 lg ( ) lg 5 (d) lg 4lg +=0 (e) 1 4 lg4 lg +8=0 (f) log +4 = 4 (g) log +log =1 9. Solve the given equations. (a) 4 1 =4 1 (b) = (c) 1 +1 = +1 (d) = (e) 6 1 = Solve the inequalities (a) ( +1)( ) 0 (b) +6 5 (c) (d) (e) +7 1

14 41. For which values of is the function ( ) =( ) + increasing? Decreasing? Constant? 4. Determine the sign of the given functions : R R. (a) ( ) = 6 +5 (b) ( ) = (c) ( ) = 5 6 (d) ( ) = + +1 (e) ( ) = Determine the sign of the given functions (also indicate the largest domain of definition of the function). (a) ( ) = + (b) ( ) = 4 +( +) 44. Determine the number of solutions of the equation ( +1) +8 = 0 in terms of the parameter R. 45. For what values of R we have ( 1) for any R? 46. For the given function, find the vertex, axis of symmetry and -intercepts (if any), then graph. (a) ( ) = 6 +8 (b) ( ) = +7 5 (c) ( ) = +6 9 (d) ( ) = +1 (e) ( ) = Solve the given equations. (a) =0 (b) 4 10 = 4 (c) = (d) p 1 4 = Determine the quadratic function ( ) = + + such that its graph passes through the points (1 ), ( 1 6) and ( ). 49. Determine the quadratic function ( ) = + + with vertex at the point (4 4) and -intercept (0 1). 50. For the given functions : R R, determine if they are (or not) increasing, decreasing, injective, surjective, bijective. If they are bijective, also find the inverse function. ½ +1 0 (a) ( ) = +1 0 ½ 1 (b) ( ) = + 1 ½ +1 0 (c) ( ) =