3.1 Basic properties of real numbers  continuation Inmum and supremum of a set of real numbers


 Jody Lang
 1 years ago
 Views:
Transcription
1 Chapter 3 Real numbers The notion of real number was introduced in section 1.3 where the axiomatic denition of the set of all real numbers was done and some basic properties of the set of all real numbers were shown. This chapter is dedicated to the properties of the set R and its subsets. These properties follows from the Dedekind cut D(R) and from the axioms of ordering. 3.1 Basic properties of real numbers  continuation Inmum and supremum of a set of real numbers Having in mind we will work with the sets of real numbers, we will give a specic characterization of the inmum and supremum, respectively, in R. THEOREM Let B R be nonempty. Then: 1. inf B = b R i (a) x B : b x; (b) ɛ > 0 x 0 B : x 0 < b + ɛ. 2. sup B = b R i (α) x B : x b; (β) ɛ > 0 x 0 B : b ɛ < x 0. Proof. We will prove the rst part of the theorem. The second part can be proven in fully analogical way. Let inf B = b R. The (a) property coincides with the rs property of an inmum mentioned in Theorem The same theorem implies from b < b + ɛ ( R) that x 0 < b + ɛ and this veries the (b) property. Now, let the properties (a) and (b) hold true. For y R such that b < y we put ɛ = y b > 0. 5
2 3.1 Basic properties of real numbers  continuation 6 Then the property (b) implies there exists x 0 in B such that x 0 < b + (y b) = b. This completes the proof of the second property of inmum from Theorem We have shown in the example that the set B = {x Q; x 2 3, x 0} is bounded from above (in Q) but has no supremum in Q. The Dedekind cut for real numbers implies that the subset in R behave essentially dierently from the subsets in Q with respect to supremum. THEOREM Any nonempty set M R that is bounded from above has a supremum in R. Proof. Let us dene two sets A and B as follows: The following relations hold true: B := {y R; x M : x < y}, A := R \ B. = A R, = B R, A B = R. We will prove that for any element a A and any element b B we have: a < b. Let us suppose there are two numbers a 0 A, b 0 B such that a 0 b 0, then for all x M we have x < b 0 a 0. Hence a 0 must be in B and this is the contradiction with the fact that A B =. We have shown that the pair of sets A, B is a Dedekind cut (see section 1.3) and therefore there is a real number c such that a c b for all a A and b B. By using Theorem we will show that c = sup M. The inclusion M A veries the (α) property for c. We have to prove that the (β) property also holds true. Let ɛ > 0. Then c ɛ < c b, b B, and therefore c ɛ / B. If for any x M would be x < c ɛ then it would be also c ɛ B and this would be in contradiction with the statement that c ɛ / B. Thus, the (β) property holds true and this means: ɛ > 0 x 0 M : c ɛ < x 0. There is, naturally, an analogical theorem about inmum: THEOREM Any nonempty set M R that is bounded from below has an inmum in R. Proof. Let k R is a lower bound of a set M R, then k is a upper bound of the set M 1 = {y R; y = x, x M}. Theorem guarantees existence of sup M 1 = c 1 R. It is obvious that inf M = c 1. 6
3 3.1 Basic properties of real numbers  continuation 7 Example Let = M 1 M R and let M be bounded from above. The sup M 1 sup M in R. Solution: The existence of the following two real numbers sup M =: b, sup M 1 =: b 1 is guaranteed by Theorem Using the denition of supremum (Denition 1.5.4) we obtain x M 1 M : x b and b 1 b. Example Show that the set of all irrational numbers R \ Q is nonempty. Solution: Example tells us that the set B = {x Q; x 2 < 3, x 0} R is bounded from above in Q, and therefore it is bounded from above in R also. However, sup B / Q. Thus, with respect to Theorem we obtain sup B R and subsequently sup B R \ Q Archimedean property By using Theorem we will show that the Archimedean property holds true for the real numbers. (Compare this with the Archimedean axiom IV Q from Denition 1.3.1). THEOREM It holds true: Proof. By contradiction. Let x R n N : x < n. y R n N : n y. This means that the set of all natural numbers N is bounded from above and therefore (Theorem 3.1.2) it exists sup N = b R. With respect to the (β) property from Theorem implies for ɛ = 1 that there exists n 0 in N such that b 1 < n 0. Subsequently, we have b < n N and this is in contradiction with that b = sup N On density of the rational numbers and the irrational numbers in R The set R, as shown in Example 3.1.2, is the extension of the set of all rational numbers Q and these two sets diers by nonempty set of all irrational numbers. We will show in the following theorem that both sets Q and R \ Q are "dense" in R. We will start with the supply lemma: 7
4 3.1 Basic properties of real numbers  continuation 8 Lemma The statement: x R n Z : n x < n + 1. The integer n is dened by x uniquely. Proof. Let x R. By Archimedean property we have n 1 N, n 2 N : x < n 1, x < n 2. Therefore n 1 < x < n 2. Let us construct the integers n 1, n 1 + 1,..., n 2 1, n 2 and denote by n the greatest of them such that n x. Such an integer is unique and it obeys n x < n + 1. Denition Let x R. The integer n such that n x < n + 1 is called the integral part of the real number x and it is denoted by [x]. 3 5 For example 2 = 0, 8 = 2. THEOREM Let a < b be a pair of real numbers. Then there exist such a rational number x and such an irrational number y that a < x < b, and a < y < b. Proof. By the Archimedean property we know there exists such a natural number n that 1 n < b a, b a > 0. Let k = [na]. Then we have k na < k + 1 and also The inequality k/n a implies k n a < k + 1 n. k + 1 n a + 1 n So we have shown that the rational number 8 < a + (b 1) = b. x = k + 1 n
5 3.1 Basic properties of real numbers  continuation 9 obeys a < x < b. We know that the number sup B from Example is irrational. Let us choose x in such a way that a sup B < x < b sup B. (The existence of such a rational x is guaranteed). Then y = x + sup B is an irrational number for which the requested inequality holds true. Example Show that for any x R we have: x = sup{u Q; u < x}. Solution: The (α) property of supremum from Theorem is obvious. Let ɛ > 0. Theorem implies that r 0 Q : x ɛ < r 0 < x. Thus, r 0 {u Q; u < x} and the (β) property of Theorem holds true Root of a real number The following theorem guarantees the existence and the uniqueness of the nth root of a positive real number. THEOREM It holds true that a > 0 n N y > 0 : y n = a. The number y is dened by the numbers a and n uniquely. Proof. Let B := {x > 0; x n a}. The set B is nonempty since b := a a + 1 B, in fact: 0 < b < 1 and bn < b < a. Furthermore, the set B is bounded from above by the number c := a + 1. In fact, if some y 0 B would obey y 0 > a + 1 then it would be yn n > (a + 1) n, and this would be in contradiction with our assumption that y 0 belongs to B. At the moment Theorem guarantees existence of R + y := sup B. This number y must obey just one of the followings: y n < a, y n > a, y n = a. The rst two cases can be excluded: 9
6 y n + h n Basic properties of real numbers  continuation 10 (1) let us suppose y n < a. We can choose a positive real number h < 1 such that a y n h < (y + 1) n y, (a n yn > 0). + Œ + Œ By making use of binomial theorem we obtain n n (y + h) n = y n y n 1 h + h n 1 n Œ y n n n Œ = y n + h [(y + 1) n y n ] < y n + (a y n ) = a. Thus we have shown that y + h B and this is the contradiction since y = sup B. (2) One can prove that y n > a cannot hold true by analogy. (3) Finally, the relation must hold true. y n = a The uniqueness can be shown indirectly. Let y 1, y 2 R + be two numbers such that Then we obtain the following contradiction This completes the proof. y n 1 = a, y n 2 = a, y 1 < y 2. a = y n 1 < y n 2 = a. Theorem allows us to write the following denition Denition Let a > 0 and n N. The number y > 0 for which y n = n is called the nth root of the number a. It us denoted by the symbol n a or by a 1/n. 2. For a = 0 we put n 0 := For a < 0 and an odd n N we dene n a = n a. It follows from the proof of Theorem and from Examples and that 3 = sup{x R + ; x 2 < 3} = sup{x Q; x 2 < 3, x 0} is an irrational number. 10
7 3.1 Basic properties of real numbers  continuation Extended set of real numbers For the further purposes in Chapter 6 we will need some extension R of the set of all real numbers R. The set R will be closed under operation of taking a limit, i.e. a mapping with values in R R will have a limit in R. Such a set can be dened with help of relation of natural ordering in R: Denition where for all x R: < x < +. R := R {, + }, For our purposes we need not to dene any operation with improper elements ±, it will be sucient to know the topological structure of the set R only (see Theorem 5.2.5). By denition and Theorems and we obtain: THEOREM Any nonempty set M R has a supremum as well as an inmum in R. For example, sup Z = +, inf Z = (in R), where sup Z and inf Z do not exist in real numbers. Problems 1. Prove in details the second part of Theorem Prove that Theorem as well as Theorem are equivalent with the Dedekind cut mentioned in the third section of the rst Chapter. 3. Let = M 1 M R and let M 1 be bounded from below. Prove that inf M 1 inf M in R. 4. Show that for any real number x we have 5. Show that x = inf{u Q : x < u} = sup{u Q : u < x}. a R, a > 1 b R + 0 n Z : a n 1 b < a n. (Hint: you can use the Archimedean property.) 11
8 3.2 Notion of interval, intervals over the real axis Let = M R and M 1 = {y R; y = x, x M}. Show that sup M 1 = inf M, inf M 1 = sup M. 7. Find supremum as well as inmum in R and in R of the following mappings, a) f : x x n x, x R \ {1}, b) =: (a n ) n N n ( 1) nšn N n, n N c1) f 1 : x [x], x {t R; 0 < t < 2}, c2) f 2 : x [x], x {t R; 0 t 2}, c3) f 3 : x x [x], x {t R; 0 t 1}. 8. Prove Theorem Answers 7 a) There are no min f, max f, inf f, sup f in R and min f = inf f = and max f = sup f = + in R. b) In the set R: do not exist and In the set R: min a n, min b n, max b n, sup n N n N n N b n n N inf a n = 1, max a n = sup a n = 2, inf b n = 0. max b n = sup b n = +. Further relations coincides with those valid in R. c) In the set R min f 1, and max f 1 do not exist, inf f 1 = 0, sup f 1 = 1; min f 2 = inf f 2 = 0, max f 2 = sup f 2 = 2; min f 3 = inf f 3 = 0, max f 3 does not exist and sup f 3 = Notion of interval, intervals over the real axis Studying continuity, dierentiability and integrability of a mapping (function) we will see that these properties do depend not only on a relation that denes the mapping but they do depend also on denition domain of the mapping. This is the reason why we will be interested in more details in most frequently used subsets of the real axis  intervals. 12
9 3.2 Notion of interval, intervals over the real axis Intervals Denition A set I R is called an interval (from R) i it has the following properties: a) The set I contains at least two distinct points, b) If x 1 I, x 2 I then ( x R : x 1 < x < x 2 ) x I. 2 A cartesian product of m intervals (from R) is called an mdimensional interval (from R). It is easy to verify that if a R, b R and if a < b then any of the following sets is an interval from R: I 1 = {x R : a x b} =: [a, b] (closed interval) I 2 = {x R : a x < b} =: [a, b) from the right etc.) I 3 = {x R : a < x b} =: (a, b] from the left etc.) (halfclosed interval or halfopen interval, open (halfclosed interval or halfopen interval, open I 1 = {x R : a < x < b} =: (a, b) (open interval) Let us remark that any of the sets I i, i = 1, 2, 3, 4 represents four intervals from R with respect to whether a, b R or a = or b = +. For example, for a, b R the interval I 1 can behave as [a, b], [a, + ], [, b], [, + ] = R. Analogically for other intervals. A natural question arises whether any interval can be identied with one of the types I i, i = 1, 2, 3, 4. The answer to this question together with full characterization of the set of all intervals is given in the following theorem. THEOREM A set I R is an interval from R if and only if there exists an index i in {1, 2, 3, 4} such that I = I i. Proof. 1. It is obvious from the text above this theorem that if I = I i, i = 1, 2, 3, 4 then I is an interval (in R). 2. Let know I be an interval. Then (Theorem 3.1.7) there exist a := inf I and b := sup I in R. Since I contains at least two distinct points, we have a < b. The denitions of inmum and supremum of the set I imply: x I : a x b. ( ) 13
10 3.2 Notion of interval, intervals over the real axis 14 Four distinct cases can appear: a) a I, b I, b) a I, b / I, c) a / I, b I, a) a / I, b / I. Let us consider case a). (In other cases the proof is similar.) Eq. ( ). implies that I [a, b]. Let x [a, b]. If x = a or x = b then x I. Let now x (a, b). Since a = inf I it exists an x 1 I such that x 1 < x (see Theorem 3.1.5). Similarly, b = sup I implies that it exists an x 2 I such that x < x 2. So we have that x 1 < x < x 2. The (b) property of the interval we have x I, thus [a, b] I. We have shown that I = [a, b]. Example Write down some examples of two dimensional intervals from R 2 and nd out what is their number. Solution: By Theorem and the second part of Denition the following sets are intervals in R 2 (a, b, c, d R, a < b, c < d): [a, b] [c, d], [a, b] [, d], [a, + ] [c, d], [a, + ] [, d], [, b] [c, d], [, b] [, d], [, + ] [c, d], [, + ] [, d], [a, b] [c, + ], [a, b] [, ], [a, + ] [c, + ], [a, + ] [, + ], [, b] [c, + ], [, b] [, + ], [, + ] [c, + ], [, + ] [, + ]. Other two dimensional intervals can be derived from I i I j, i, j = 1, 2, 3, 4 such that i+j 2. This means we have 4 4 = 256 distinct two dimensional intervals in R Embedded intervals Denition Let the sequences of real numbers (a n ) n N and (b n ) n N obey: Then the sequence of closed intervals in R a 1 a 2 a n b n b 2 b 1. ([a n, b n ] n N ) is called the system of embedded intervals in R. For any system of embedded intervals we have: [a n+1, b n+1 ] [a n, b n ], (n N). 14
11 3.2 Notion of interval, intervals over the real axis 15 THEOREM Let ([a n, b n ] n N ) be \ a system of embedded intervals in R. Then i.e. c R n N : c [a n, b n ]. [a n, b n ], n N Proof. The sequence (a n ) n N is bounded from above by any of the numbers b n and therefore (see Theorem 3.1.2) we know that it exists the number sup(a n ) n N =: a R and a b n. Boundedness from below of the sequence (b n ) n N by a implies that there exists inf(b n ) n N =: b R (see Theorem 3.1.3) and furthermore a n a b b n, for all n N. Thus we have shown that at least a belongs to any interval of the system ([a n, b n ] n N ). The intersection \ n N 1 n, 1 n equals to onepoint set {0}. The intersection \ of the embedded system of open intervals n N0, 1 n is empty (but this fact is not in contradiction with statement of Theorem 3.2.2) An example of uncountable set THEOREM The interval [0, 1] is uncountable set. Proof. The proof will be done indirectly. Let us suppose the interval in question [0, 1] is an countable set. Then it follows from Theorem that this interval is the set of values of an injective sequence of the elements of R. Thus, there exists a sequence (c n ) n N, c n R 3 such that its values coincides with [0, 1]. Let us divide the interval [0, 1] into three parts: 0, 1 1, 3 3, 2 2, 3, 1. Then c 1 [0, 1] does not belong to at least one of these parts of the interval [0, 1]. Let us denote it by [a 1, b 1 ], so c 1 / [a 1, b 1 ]. This interval can be also divided into three parts analogically as we have done with [0, 1] one step before, i.e.: a 1, b 1 a 1 3, b1 a 1 3, 2(b 1 a 1 ) 3 1 2(b1 a 1 ),, b. 3 15
12 3.2 Notion of interval, intervals over the real axis 16 The one of the above written intervals which is such that c 2 does not belong to it will be denoted as [a 2, b 2 ], obviously [a 2, b 2 ] [a 1, b 1 ]. By induction we can construct the sequence of embedded closed intervals in R with property: n N : c n / [a n, b n ]. By making use of Theorem we know that Since c [0, 1] we have: and this contradicts the previous statement Real line c R n N : c [a n, b n ]. n 0 N : c = c n0 / [a n0, b n0 ], In order to be able to interpret geometrically various fact and properties of mappings we will give a description of relation between the set of all real numbers R and the set of points of a line. This relation will be described using the real line axiom. We have already used this axiom in an intuitive way. Let us denote by ρ(a, B) the length of the line segment with endpoints A and B. (We suppose we can measure the lengths of line segments.) Let us have two distinct points O and J on the line o. Let us suppose ρ(o, J) = 1. Then we can dene a mapping: in the following way: f : R o (α) For x 1 R + we dene f(x 1 ) =: P x o is such a point on a halfline OJ that ρ(o, P x ) = x 1 (β) For x 2 R we dene f(x 1 ) =: Q x o is such a point on a halfline OJ that ρ(o, Q x ) = x 2 (γ) For x = 0 we dene f(x) = O. This situation is shown on Figure 3.1. Real line axiom The mapping f : R o with properties (α), (β), (γ) is bijective. 16
13 3.2 Notion of interval, intervals over the real axis 17 x x1 Q x O J P x Figure 3.1: o The line o which is with respect to Real line axiom the image of the set R under the mapping f is called real line, its point O is called the beginning and the line segment OJ is called the unity. One can dene on the real line an inner binary relation of addition of two points A, B o (we denote it by the symbol ) and an external operation of multiplication of a point A o by a number s R (we denote it by the symbol ) as follows: A B =: E o, s A =: F o, ( ) where the point E is obtained as the end point of the addition of oriented segments OA and OB (see Figure 3.2). The point O is the beginning of the real line o and ρ(o, B) = ρ(o, B ). o O J A = O B B Figure 3.2: The point F can be constructed by homothety as shown in Figure 3.3. The point S o is an image f(s) (f is the mapping from the real line axiom) and p o is a line crossing through the beginning O of the real line o. F p S O J S A F o Figure 3.3: The following theorem holds true. THEOREM The vector space of the points of real line over R (o, R;,, =) is isomorphic with the vector space of the real numbers over R (R, R; +,, =). 17
14 3.2 Notion of interval, intervals over the real axis 18 Proof. It is easy to nd out that (o, R;,, =) is in fact the vector space. Let us consider the bijective mapping f dened in the real line axiom. For a, b R we put f(a) =: A, f(b) =: B. Then we obtain by the properties (α), (β), (γ) and ( ) that f(a + b) = A B = f(a) f(b), f(sa) = s (A) = s f(a). This theorem allows for considering the set of real numbers R and the real line o as equivalent from the algebraic point of view. Furthermore, we can write a = A(= f(a)) for any a R and we can use the real line o as a geometrical tool for representation of the real numbers. Problems 1. Let I 1, I 2 R be intervals. Decide when the sets are intervals. I 1 I 2, I 1 I 2 2. Write down few patterns of two dimensional intervals from R 2 and nd out their number. 3. Prove the following: A set I R is an interval if and only if it can be written as one of the next nine possibilities (a, b, c, d R, a < b): [a, b], [a, b), (a, b], (a, b), [c, + ), (c, + ), (, d], (, d), (, + ). (Hint: see the proof of Theorem 3.2.1) 4. Prove the following: for any a, b R, a < b the intervals [a, b], (a, b) and R are uncountable sets. (Hint: the function dened by f : [0, 1] [a, b] f : x (b a)x + a is bijective. To prove that the set R is uncountable you can use the function ) 5. Prove Theorem in details. g : (0, 1) R, g(x) = 1 x + 1 x 1. 18
15 3.3 Further extensions of the real numbers 19 Answers 1 The set I 1 I 2 is an interval i I 1 I 2. The set I 1 I 2 is an interval i it contains at least two distinct points Further extensions of the real numbers The extension of the set of all rational numbers to the set of real numbers was motivated by practical purposes, e.g. to give sense to the solution to the algebraic equation: x 2 = 3. Another question is related to the fact that we have not dened even root of a negative number (see Denition 3.1.2). The equation x 2 = 3 has no solution in R. However, high school knowledge tells us that this equation can be solved in the set of complex numbers C. And this set C is, in some sense, also an extension of the set of all real numbers. In this section we will remind basic properties of the set of all complex numbers Field of complex numbers Since the set R is isomorphic with the set of all pairs R {0} (an isomorphism is given by the mapping: f : x (x, 0), x R), the following denition will be a reasonable generalization of the set of all real numbers. Denition The set of all complex numbers C is the cartesian product R R on which we have dened an inner binary operation of addition and multiplication (the equivalence of pairs is dened in the fourth section of the rst Chapter) in this way: 1 Let z 1 = (x 1, y 1 ) C, z 2 = (x 2, y 2 ) C. Then the addition of the complex numbers z 1 and z 2 is the complex number: z 1 + z 2 = (x 1 + x 2, y 1 + y 2 ). 2 The product of z 1 and z 2 is the complex number: z 1 z 2 = (x 1 x 2 y 1 y 2, x 1 y 2 + x 2 y 1 ). 19
16 3.3 Further extensions of the real numbers 20 The following theorem is a simple consequence of previous denition: THEOREM The set of all complex numbers taken as (C, +,, =) is a eld and taken as (C, R, +,, =) is a vector space over R. Proof. It suces to verify the properties of eld and vector space, respectively (they are introduced in Denitions and 1.6.5). The external operation  product " " of a vector z = (x, y) C with a scalar s R (the set {(x, 0); x R} is isomorphic with R) in the vector space (C, R, +,, =) is dened by: sz = (s, 0) (x, y) = (sx, sy) C. The neutral element with respect to addition is 0 := (0, 0) C. The neutral element with respect to multiplication is (1, 0) C. The inverse element (with respect to addition) to an element z = (x, y) C is z = ( x, y). The inverse element (with respect to multiplication) to an element z = (x, y) is dened if z 0 and is denoted by z 1 (x 0, y 0 ) and is given by the condition z 1 z = (1, ) (xx 0 yy 0, xy 0 + yx 0 ) = (1, 0). This system of equation has unique solution that reads Thus, x 0 = z 0 = x x 2 + y, y 2 0 = y x 2 2Œ + y. 2 x x 2 + y, y. 2 x 2 + y If z = (x, y) C then we dene: x =: Rez, y =: Imz and we call them the real part and the imaginary part of the complex number z. If we denote complex number (0, 1) as i ((0, 1) =: i, the imaginary unit) then and Complex number i 2 = ( 1, 0) = 1, z = (x, y) = (x, 0) + (0, y) = (x, 0) + (0, 1)(y, 0) =: x + iy. z = (x, y) = x iy 20
17 3.3 Further extensions of the real numbers 21 is called the complex conjugate to the complex number z = (x, y) = x + iy. It is obvious that z z = x 2 + y 2, z + z = 2x, x = 1 (z + z), 2 y = 1 (z z). 2i Complex plane The geometrical interpretation of complex numbers is given in the following theorem THEOREM Let o 1, o 2 be two real axes with common beginning. Then the vector space of complex numbers (C, R; +,, =) is isomorphic with the vector space (o 1 o 2, R,,, =). An inner binary operation means the vector addition and the external binary operation represents the multiplication of a vector by a real scalar. Proof. With help of bijective mapping f introduced in Axiom we can dene an bijection from R 2 onto o 1 o 2 : by Figure 3.4. g(x, y) = Z o 1 o 2, (x, y) R R, y =: B o 2 Z O x =: A o 1 Figure 3.4: Denitions of the operations and can be also easily formulated by Figures 3.5 and 3.6. Let (x 1, y 1 ), (x 2, y 2 ) R 2 and let f 1, f 2 be a bijection from R onto o 1 and o 2, respectively, such that f 1 (x i ) = A i, f 2 (y i ) = B i, i = 1, 2. 21
18 3.3 Further extensions of the real numbers 22 o 2 B 1 B 2 S 1 y 2 =: B 2 Z 2 y 1 =: B 1 Z 1 O x 2 =: A 2 x 1 =: A 1 A 1 A 2 o 1 Figure 3.5: Let then g(x 1, y 1 ) = Z 1 o 1 o 2, g(x 2, y 2 ) = Z 2 o 1 o 2, s R, Z 1 Z 2 = S 1 o 1 o 2, sz 1 = S 2 o 1 o 2. The mapping g is then the requested isomorphism. If the axes o 1 and o 2 from Theorem are perpendicular then their cartesian product o 1 o 2 is called the complex plane. The axis o 1 is called real axis and the axis o 2 is called the imaginary axis. It is exactly Theorem that allows for considering the set of all complex numbers C and the complex plane to be equal from the algebraic point of view. We will write for any (x, y) C. (x, y) = Z(= g(x, y)), Polar form of a complex number From the above mentioned geometrical interpretation of a complex number one easily obtains the following polar form of a complex number. THEOREM For any z = (x, y) C, z 0 there exists just one φ [0, 2π)  called the argument or the phase of the complex number z  and just one r > 0  called the absolute 22
19 3.3 Further extensions of the real numbers 23 o 2 sb 1 =: E S 2 S y 1 =: B 1 Z 1 O J x 1 =: A 1 s =: S sa1 =: D o 1 Figure 3.6: value or the modulus of the complex number z  such that Proof. The proof is evident from Figure 3.7. z = (r cos(φ), r sin(φ)) = r(cos(φ) + i sin(φ)). o 2 y Z r φ x o 1 It holds true that the numbers: Figure 3.7: r =È x2 + y 2 R +, φ [0, 2π) 23
20 3.3 Further extensions of the real numbers 24 are dened uniquely by equations: x = r cos(φ), y = r sin(φ). The number r is often denoted by z. An exact denition of used goniometric functions is given in section 4.4 of this book. Problems 1. Prove in details Theorems and Formulate and explain the algorithm of construction of the image of: a) product of complex numbers b) symmetric element z 1 to an element z = (x, y) C in the complex plane by using the isomorphism g from Theorem Show that for z 1 = r 1 (cos(φ 1 ) + i sin(φ 1 )) C and z 2 = r 2 (cos(φ 2 ) + i sin(φ 2 )) C, for any n N we have: a) z 1 z 2 = r 1 r 2 (cos(φ 1 + φ 2 ) + i sin(φ 1 + φ 2 )), b) z n 1 = r n (cos(nφ) + i sin(nφ))  Moivre theorem, c) n z1 = z C; z = n r cos φ + 2kπ n Œ + i sin φ + 2kπ n Œ, for k = 0, 1, 2,..., n 1«, (let us remind that n z 1 is a set of all complex numbers z for which z n = z 1 ), d) z 1 0 and z 1 = 0 z 1 = 0 C, e) z 1 z 2 = z 1 z 2, f) z 1 z 2 z 1 ± z 2 z 1 + z 2. 24
Introduction to Real Analysis
Christopher Heil Introduction to Real Analysis Chapter 0 Online Expanded Chapter on Notation and Preliminaries Last Updated: January 9, 2018 c 2018 by Christopher Heil Chapter 0 Notation and Preliminaries:
More informationChapter One. The Real Number System
Chapter One. The Real Number System We shall give a quick introduction to the real number system. It is imperative that we know how the set of real numbers behaves in the way that its completeness and
More information2.2 Some Consequences of the Completeness Axiom
60 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.2 Some Consequences of the Completeness Axiom In this section, we use the fact that R is complete to establish some important results. First, we will prove that
More informationStructure of R. Chapter Algebraic and Order Properties of R
Chapter Structure of R We will reassemble calculus by first making assumptions about the real numbers. All subsequent results will be rigorously derived from these assumptions. Most of the assumptions
More informationPrinciples of Real Analysis I Fall I. The Real Number System
21355 Principles of Real Analysis I Fall 2004 I. The Real Number System The main goal of this course is to develop the theory of realvalued functions of one real variable in a systematic and rigorous
More information11691 Review Guideline Real Analysis. Real Analysis.  According to Principles of Mathematical Analysis by Walter Rudin (Chapter 14)
Real Analysis  According to Principles of Mathematical Analysis by Walter Rudin (Chapter 14) 1 The Real and Complex Number Set: a collection of objects. Proper subset: if A B, then call A a proper subset
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need
More informationGeneral Notation. Exercises and Problems
Exercises and Problems The text contains both Exercises and Problems. The exercises are incorporated into the development of the theory in each section. Additional Problems appear at the end of most sections.
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend
More informationPOL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005
POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1
More informationEconomics 204 Summer/Fall 2011 Lecture 2 Tuesday July 26, 2011 N Now, on the main diagonal, change all the 0s to 1s and vice versa:
Economics 04 Summer/Fall 011 Lecture Tuesday July 6, 011 Section 1.4. Cardinality (cont.) Theorem 1 (Cantor) N, the set of all subsets of N, is not countable. Proof: Suppose N is countable. Then there
More informationDue date: Monday, February 6, 2017.
Modern Analysis Homework 3 Solutions Due date: Monday, February 6, 2017. 1. If A R define A = {x R : x A}. Let A be a nonempty set of real numbers, assume A is bounded above. Prove that A is bounded below
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationReal Analysis  Notes and After Notes Fall 2008
Real Analysis  Notes and After Notes Fall 2008 October 29, 2008 1 Introduction into proof August 20, 2008 First we will go through some simple proofs to learn how one writes a rigorous proof. Let start
More informationMathematics 220 Workshop Cardinality. Some harder problems on cardinality.
Some harder problems on cardinality. These are two series of problems with specific goals: the first goal is to prove that the cardinality of the set of irrational numbers is continuum, and the second
More informationSets, Structures, Numbers
Chapter 1 Sets, Structures, Numbers Abstract In this chapter we shall introduce most of the background needed to develop the foundations of mathematical analysis. We start with sets and algebraic structures.
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationChapter 1. Sets and Mappings
Chapter 1. Sets and Mappings 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationREAL ANALYSIS: INTRODUCTION
REAL ANALYSIS: INTRODUCTION DR. RITU AGARWAL EMAIL: RAGARWAL.MATHS@MNIT.AC.IN MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY JAIPUR Contents 1. The real number system 1 2. Field Axioms 1 3. Order Axioms 2 4.
More informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationChapter 1 The Real Numbers
Chapter 1 The Real Numbers In a beginning course in calculus, the emphasis is on introducing the techniques of the subject;i.e., differentiation and integration and their applications. An advanced calculus
More informationMATH 102 INTRODUCTION TO MATHEMATICAL ANALYSIS. 1. Some Fundamentals
MATH 02 INTRODUCTION TO MATHEMATICAL ANALYSIS Properties of Real Numbers Some Fundamentals The whole course will be based entirely on the study of sequence of numbers and functions defined on the real
More informationIntroduction to Mathematical Analysis I. Second Edition. Beatriz Lafferriere Gerardo Lafferriere Nguyen Mau Nam
Introduction to Mathematical Analysis I Second Edition Beatriz Lafferriere Gerardo Lafferriere Nguyen Mau Nam Introduction to Mathematical Analysis I Second Edition Beatriz Lafferriere Gerardo Lafferriere
More informationStudying Rudin s Principles of Mathematical Analysis Through Questions. August 4, 2008
Studying Rudin s Principles of Mathematical Analysis Through Questions Mesut B. Çakır c August 4, 2008 ii Contents 1 The Real and Complex Number Systems 3 1.1 Introduction............................................
More informationEconomics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011
Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Section 2.6 (cont.) Properties of Real Functions Here we first study properties of functions from R to R, making use of the additional structure
More informationA NEW SET THEORY FOR ANALYSIS
Article A NEW SET THEORY FOR ANALYSIS Juan Pablo Ramírez 0000000249122952 Abstract: We present the real number system as a generalization of the natural numbers. First, we prove the cofinite topology,
More informationPostulate 2 [Order Axioms] in WRW the usual rules for inequalities
Number Systems N 1,2,3,... the positive integers Z 3, 2, 1,0,1,2,3,... the integers Q p q : p,q Z with q 0 the rational numbers R {numbers expressible by finite or unending decimal expansions} makes sense
More informationA LITTLE REAL ANALYSIS AND TOPOLOGY
A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set
More informationSequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.
Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence
More informationChapter 1. Sets and Numbers
Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationAdvanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010
Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with
More informationEcon Lecture 2. Outline
Econ 204 2010 Lecture 2 Outline 1. Cardinality (cont.) 2. Algebraic Structures: Fields and Vector Spaces 3. Axioms for R 4. Sup, Inf, and the Supremum Property 5. Intermediate Value Theorem 1 Cardinality
More informationSuppose R is an ordered ring with positive elements P.
1. The real numbers. 1.1. Ordered rings. Definition 1.1. By an ordered commutative ring with unity we mean an ordered sextuple (R, +, 0,, 1, P ) such that (R, +, 0,, 1) is a commutative ring with unity
More informationLogical Connectives and Quantifiers
Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationMATH NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS
MATH. 4433. NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS TOMASZ PRZEBINDA. Final project, due 0:00 am, /0/208 via email.. State the Fundamental Theorem of Algebra. Recall that a subset K
More informationS15 MA 274: Exam 3 Study Questions
S15 MA 274: Exam 3 Study Questions You can find solutions to some of these problems on the next page. These questions only pertain to material covered since Exam 2. The final exam is cumulative, so you
More informationTheorems. Theorem 1.11: GreatestLowerBound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: th Root of Real Numbers
Page 1 Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: GreatestLowerBound Property Suppose is an ordered set with the leastupperbound property Suppose, and is bounded below be the set of lower
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More informationFinal Exam Review. 2. Let A = {, { }}. What is the cardinality of A? Is
1. Describe the elements of the set (Z Q) R N. Is this set countable or uncountable? Solution: The set is equal to {(x, y) x Z, y N} = Z N. Since the Cartesian product of two denumerable sets is denumerable,
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationAnalysis I. Classroom Notes. H.D. Alber
Analysis I Classroom Notes HD Alber Contents 1 Fundamental notions 1 11 Sets 1 12 Product sets, relations 5 13 Composition of statements 7 14 Quantifiers, negation of statements 9 2 Real numbers 11 21
More informationAdvanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017
Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with
More informationMathematical Reasoning & Proofs
Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons AttributionShareAlike 4.0
More informationNumber Axioms. P. Danziger. A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b. a b S.
Appendix A Number Axioms P. Danziger 1 Number Axioms 1.1 Groups Definition 1 A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b and c S 0. (Closure) 1. (Associativity)
More informationMATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017
MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A
More informationIntroduction to Real Analysis
Introduction to Real Analysis Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 1 Sets Sets are the basic objects of mathematics. In fact, they are so basic that
More informationLecture Notes in Real Analysis Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay
Lecture Notes in Real Analysis 2010 Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay August 6, 2010 Lectures 13 (Iweek) Lecture 1 Why real numbers? Example 1 Gaps in the
More informationREAL VARIABLES: PROBLEM SET 1. = x limsup E k
REAL VARIABLES: PROBLEM SET 1 BEN ELDER 1. Problem 1.1a First let s prove that limsup E k consists of those points which belong to infinitely many E k. From equation 1.1: limsup E k = E k For limsup E
More informationDefinition: Let S and T be sets. A binary relation on SxT is any subset of SxT. A binary relation on S is any subset of SxS.
4 Functions Before studying functions we will first quickly define a more general idea, namely the notion of a relation. A function turns out to be a special type of relation. Definition: Let S and T be
More informationConsequences of the Completeness Property
Consequences of the Completeness Property Philippe B. Laval KSU Today Philippe B. Laval (KSU) Consequences of the Completeness Property Today 1 / 10 Introduction In this section, we use the fact that R
More informationEntrance Exam, Real Analysis September 1, 2017 Solve exactly 6 out of the 8 problems
September, 27 Solve exactly 6 out of the 8 problems. Prove by denition (in ɛ δ language) that f(x) = + x 2 is uniformly continuous in (, ). Is f(x) uniformly continuous in (, )? Prove your conclusion.
More informationIntroduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION
Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from
More informationTHE REAL NUMBERS Chapter #4
FOUNDATIONS OF ANALYSIS FALL 2008 TRUE/FALSE QUESTIONS THE REAL NUMBERS Chapter #4 (1) Every element in a field has a multiplicative inverse. (2) In a field the additive inverse of 1 is 0. (3) In a field
More informationMATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 2: Countability and Cantor Sets
MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 2: Countability and Cantor Sets Countable and Uncountable Sets The concept of countability will be important in this course
More informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics and Engineering Lecture notes for PDEs Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 The integration theory
More informationChapter 1 Preliminaries
Chapter 1 Preliminaries 1.1 Conventions and Notations Throughout the book we use the following notations for standard sets of numbers: N the set {1, 2,...} of natural numbers Z the set of integers Q the
More informationCHAPTER 8: EXPLORING R
CHAPTER 8: EXPLORING R LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN In the previous chapter we discussed the need for a complete ordered field. The field Q is not complete, so we constructed
More informationNotes on Complex Analysis
Michael Papadimitrakis Notes on Complex Analysis Department of Mathematics University of Crete Contents The complex plane.. The complex plane...................................2 Argument and polar representation.........................
More informationUNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE. B.Sc. MATHEMATICS V SEMESTER. (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE B.Sc. MATHEMATICS V SEMESTER (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS QUESTION BANK 1. Find the number of elements in the power
More informationA Short Review of Cardinality
Christopher Heil A Short Review of Cardinality November 14, 2017 c 2017 Christopher Heil Chapter 1 Cardinality We will give a short review of the definition of cardinality and prove some facts about the
More informationThat is, there is an element
Section 3.1: Mathematical Induction Let N denote the set of natural numbers (positive integers). N = {1, 2, 3, 4, } Axiom: If S is a nonempty subset of N, then S has a least element. That is, there is
More information1 The Real Number System
1 The Real Number System The rational numbers are beautiful, but are not big enough for various purposes, and the set R of real numbers was constructed in the late nineteenth century, as a kind of an envelope
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationContents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3
Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability
More informationLebesgue Measure. Dung Le 1
Lebesgue Measure Dung Le 1 1 Introduction How do we measure the size of a set in IR? Let s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its
More informationSome Background Material
Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary  but important  material as a way of dipping our toes in the water. This chapter also introduces important
More information2. Two binary operations (addition, denoted + and multiplication, denoted
Chapter 2 The Structure of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference between
More informationPart IA Numbers and Sets
Part IA Numbers and Sets Definitions Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationNotes on ordinals and cardinals
Notes on ordinals and cardinals Reed Solomon 1 Background Terminology We will use the following notation for the common number systems: N = {0, 1, 2,...} = the natural numbers Z = {..., 2, 1, 0, 1, 2,...}
More informationWeek 2: Sequences and Series
QF0: Quantitative Finance August 29, 207 Week 2: Sequences and Series Facilitator: Christopher Ting AY 207/208 Mathematicians have tried in vain to this day to discover some order in the sequence of prime
More informationIntroduction to Topology
Introduction to Topology Randall R. Holmes Auburn University Typeset by AMSTEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about
More informationDefinitions & Theorems
Definitions & Theorems Math 147, Fall 2009 December 19, 2010 Contents 1 Logic 2 1.1 Sets.................................................. 2 1.2 The Peano axioms..........................................
More informationSupremum and Infimum
Supremum and Infimum UBC M0 Lecture Notes by Philip D. Loewen The Real Number System. Work hard to construct from the axioms a set R with special elements O and I, and a subset P R, and mappings A: R R
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationABSTRACT INTEGRATION CHAPTER ONE
CHAPTER ONE ABSTRACT INTEGRATION Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Suggestions and errors are invited and can be mailed
More informationAN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES
AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov
More informationNotes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.
Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3
More informationMATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7
MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7 Real Number Summary of terminology and theorems: Definition: (Supremum & infimum) A supremum (or least upper bound) of a nonempty
More informationHilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space.
Hilbert Spaces Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Vector Space. Vector space, ν, over the field of complex numbers,
More informationDescribing the Real Numbers
Describing the Real Numbers Anthony Várilly Math 25a, Fall 2001 1 Introduction The goal of these notes is to uniquely describe the real numbers by taking certain statements as axioms. This exercise might
More informationWalker Ray Econ 204 Problem Set 3 Suggested Solutions August 6, 2015
Problem 1. Take any mapping f from a metric space X into a metric space Y. Prove that f is continuous if and only if f(a) f(a). (Hint: use the closed set characterization of continuity). I make use of
More information5 Set Operations, Functions, and Counting
5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the nonnegative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,
More informationTopology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:
Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA Email address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework
More informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More information1.3. The Completeness Axiom.
13 The Completeness Axiom 1 13 The Completeness Axiom Note In this section we give the final Axiom in the definition of the real numbers, R So far, the 8 axioms we have yield an ordered field We have seen
More informationMetric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y)
Metric Space Topology (Spring 2016) Selected Homework Solutions HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) d(z, w) d(x, z) + d(y, w) holds for all w, x, y, z X.
More information(a) We need to prove that is reflexive, symmetric and transitive. 2b + a = 3a + 3b (2a + b) = 3a + 3b 3k = 3(a + b k)
MATH 111 Optional Exam 3 lutions 1. (0 pts) We define a relation on Z as follows: a b if a + b is divisible by 3. (a) (1 pts) Prove that is an equivalence relation. (b) (8 pts) Determine all equivalence
More informationMATH 202B  Problem Set 5
MATH 202B  Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there
More informationConnectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).
Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of wellknown results.
More informationChapter 2. Real Numbers. 1. Rational Numbers
Chapter 2. Real Numbers 1. Rational Numbers A commutative ring is called a field if its nonzero elements form a group under multiplication. Let (F, +, ) be a filed with 0 as its additive identity element
More informationAxioms for the Real Number System
Axioms for the Real Number System Math 361 Fall 2003 Page 1 of 9 The Real Number System The real number system consists of four parts: 1. A set (R). We will call the elements of this set real numbers,
More informationCopyright c 2007 Jason Underdown Some rights reserved. statement. sentential connectives. negation. conjunction. disjunction
Copyright & License Copyright c 2007 Jason Underdown Some rights reserved. statement sentential connectives negation conjunction disjunction implication or conditional antecedant & consequent hypothesis
More informationCONSTRUCTION OF THE REAL NUMBERS.
CONSTRUCTION OF THE REAL NUMBERS. IAN KIMING 1. Motivation. It will not come as a big surprise to anyone when I say that we need the real numbers in mathematics. More to the point, we need to be able to
More information