Real Numbers Axioms PDF

CHAPTER 3: REAL NUMBERS §1 Real numbers axiomatics. In any mathematical theory, there are a number of initial concepts that cannot be defined through other concepts. With the help of the initial concepts, several statements are formed, to which the meanings of truth are attributed. They are ca...

CHAPTER 3: REAL NUMBERS §1 Real numbers axiomatics. In any mathematical theory, there are a number of initial concepts that cannot be defined through other concepts. With the help of the initial concepts, several statements are formed, to which the meanings of truth are attributed. They are called axioms. We give the axioms of real numbers. I. Axioms for addition. Each pair of real numbers 𝑎 and 𝑏 is mapped to a number 𝑎 + 𝑏, called sum, so that 1.1 ∀𝑎 ∈ R, ∀𝑏 ∈ R, ∀𝑐 ∈ R (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) (Associativity); 1.2 ∃ element 0 ∈ R: ∀𝑎 ∈ R 𝑎 + 0 = 0 + 𝑎 = 𝑎 (the existence of a neutral element); 1.3 ∀𝑎 ∈ R ∃𝑏 ∈ R: 𝑎 + 𝑏 = 0 (the existence of an opposite element, this element is denoted by −𝑎); 1.4 ∀𝑎 ∈ R, ∀𝑏 ∈ R 𝑎 + 𝑏 = 𝑏 + 𝑎 (commutativity). II. Axioms for multiplication. Each pair of real numbers 𝑎 and 𝑏 is mapped to a number 𝑎 · 𝑏, called product, such that 2.1 ∀ 𝑎, 𝑏, 𝑐 ∈ R (𝑎 · 𝑏) · 𝑐 = 𝑎 · (𝑏 · 𝑐) (Multiplication is associative); 2.2 ∃ 1 ∈ R : ∀𝑎 ∈ R1 · 𝑎 = 𝑎 · 1 = 𝑎; 2.3 ∀𝑎 ∈ R∃𝑏 ∈ R : 𝑎 · 𝑏 = 1 (existence of the inverse element); 2.4 ∀ 𝑎, 𝑏 ∈ R𝑎 · 𝑏 = 𝑏 · 𝑎 (Multiplication is commutative). III. Axioms of order. The set R has a relation called the order relation and denoted by 6, such that¶ 3.1 ∀𝑎 ∈ R𝑎 6 𝑎; 3.2 ∀ 𝑎, 𝑏 ∈ R(𝑎 6 𝑏) или (𝑏 6 𝑎); 3.3 ∀ 𝑎, 𝑏, 𝑐 ∈ R(𝑎 6 𝑏) и (𝑏 6 𝑐) ⇒ 𝑎 6 𝑐 (transitivity); 3.4 ∀ 𝑎, 𝑏 ∈ R (𝑎 6 𝑏) и (𝑏 6 𝑎) ⇒ 𝑎 = 𝑏 (antisymmetry). IV. Axioms of communication. 4.1 ∀ 𝑎, 𝑏, 𝑐 ∈ R (𝑎 + 𝑏) · 𝑐 = 𝑎 · 𝑐 + 𝑏 · 𝑐 (distributivity); 4.2 ∀ 𝑎, 𝑏, 𝑐 ∈ R 𝑎 6 𝑏 ⇒ 𝑎 + 𝑐 6 𝑏 + 𝑐; 4.3 ∀ 𝑎, 𝑏 ∈ R (0 6 𝑎) и (0 6 𝑏) ⇒ 0 6 𝑎 · 𝑏. V. The axiom of completeness (continuity). IF 𝑋 and 𝑌 are two non-empty sets, such that ∀𝑥 ∈ 𝑋 and ∀𝑦 ∈ 𝑌 𝑥 6 𝑦, then ∃𝑐 ∈ R : 𝑥 6 𝑐 6 𝑦. Definition 1. The difference between the numbers 𝑎 and 𝑏 ∈ R is a number 𝑥 such that 𝑥 + 𝑏 = 𝑎. Definition 2. The quotient of the numbers 𝑎 and 𝑏 ̸= 0 ∈ R is a number 𝑥 such that 𝑥 · 𝑏 = 𝑎. §2 Sets N, Z, Q Definition 3. The set N of natural numbers is a subset of the set R, such that the first number is 1, and every next number, starting from the second, is obtained by addition the previous one with a unit. N = { 1, 2, 3,... }. Definition 4. Two sets 𝐴 and 𝐵 are called equivalent (equally powerful ) if there is at least one one-to-one mapping from one set to another. Definition 5. Any set equivalent to the set of natural numbers N is called countable. If the set 𝐴 is countable, then its elements can be numbered. Sets consisting of a finite number of elements are called finite. A set that is not finite is called infinite. If 𝐴 is finite a set, then the number of its elements is denoted by |𝐴| or dim 𝐴 and is called power of the set 𝐴. Definition 6. The union of natural numbers, numbers opposite to them and zero makes up the set¶ integers Z: Z = {... , −3, −2, −1, 0, 1, 2, 3,...}. Definition 7. The set of numbers of the form 𝑛𝑝 , where 𝑝 ∈ Z; 𝑛 ∈ N, is called set of rational numbers Q: {︁ 𝑝 }︁ Q = 𝑞 = | 𝑝 ∈ Z, 𝑛 ∈ N. 𝑛 Numbers that are representable as an infinite non-periodic decimal fraction are called irrational. These numbers cannot be represented as a relation of numbers 𝑛𝑝 , where 𝑝 ∈ Z, 𝑛 ∈ N. √ √ Examples. The numbers 2, 3, 𝜋, lg 2, lg 3, sin 20∘ are irrational numbers. §3 Exact Bounds of numerical sets Upper and Lower Bounds of numerical sets. Let 𝑋 ⊂ R. Definition 8. A set 𝑋 is called bounded above if ∃𝑐 ∈ R : ∀𝑥 ∈ 𝑋, 𝑥 6 𝑐. The number 𝑐 is called the upper bound of the set 𝑋. Definition 9. A set 𝑋 is said to be bounded below if ∃𝑏 ∈ R : ∀𝑥 ∈ 𝑋, 𝑥 > 𝑏. The number 𝑏 is called the lower bound of the set 𝑋. Definition 10. A set 𝑋 is called bounded if it is bounded both above and below ∃𝑐, 𝑏 ∈ R : ∀𝑥 ∈ 𝑋, 𝑏 6 𝑥 6 𝑐. Statement The set 𝑋 is bounded if and only if ∃𝑎 ∈ R : ∀𝑥 ∈ 𝑋, |𝑥| 6 𝑎. Definition 11. The maximum element of a set 𝑋 is a number 𝑎 such that 𝑎 ∈ 𝑋 : ∀𝑥 ∈ 𝑋, 𝑥 6 𝑎. Definition 12. The minimal element of a set 𝑋 is a number 𝑎 such that 𝑎 ∈ 𝑋 : ∀𝑥 ∈ 𝑋, 𝑥 > 𝑎. Definition 13. A set 𝑋 is called unbounded above if¶ ∀𝑐 ∈ R : ∃𝑥 ∈ 𝑋 : 𝑥 > 𝑐. Definition 14. A set 𝑋 is called unbounded below if ∀𝑏 ∈ R : ∃𝑥 ∈ 𝑋 : 𝑥 < 𝑏. Definition 15. A set 𝑋 is called unbounded if it is not bounded above or bounded below. Defining exact upper and lower bounds. Definition 16. The exact upper bound is the smallest of all the upper bounds, that is 𝑀 = sup 𝑋(supreme), if 1) ∀𝑥 ∈ 𝑋, 𝑥 6 𝑀 ; 2) ∀𝑀 1 < 𝑀 ∃𝑥 ∈ 𝑋 : 𝑥 > 𝑀 1. or 𝑀 = sup 𝑋 = min{𝑐 ∈ R : ∀𝑥 ∈ 𝑋, 𝑥 6 𝑐}. Definition 17. The exact lower bound is the largest of all lower bounded, that is 𝑚 = inf 𝑋(infinum), if 1) ∀𝑥 ∈ 𝑋, 𝑥 > 𝑚; 2) ∀𝑚1 > 𝑚∃𝑥 ∈ 𝑋 : 𝑥 < 𝑚1. or 𝑚 = inf 𝑋 = max{𝑏 ∈ R : ∀𝑥 ∈ 𝑋, 𝑥 > 𝑏}. Note 1. A set may not have a maximum element, but have an exact upper bound. For example, this is the set (0; 1). Note 2. If there is a maximal element of the set 𝑋, then it coincides with sup 𝑋. The examples. 1) Let 𝐴 = [2; 8], then 𝑚 = inf 𝐴 = 2, 𝑀 = sup 𝐴 = 8. 2) Let Z+ the set of all non-negative integers. Then 𝑚 = inf{𝑝| 𝑝 ∈ Z+ } = 0, 𝑀 = sup{𝑝| 𝑝 ∈ Z+ } = +∞. Theorems on the existence of exact bounds of a number set. Theorem 18. (About the existence of supremum) If a non-empty set 𝑋 is bounded above, then it has, and a unique, exact upper bound. Theorem 19. Archimedes’ principle. The set N is not bounded above. Доказательство. Допустим, N— ограничено сверху. Proof. Suppose N is bounded above. Then it has an exact upper bound. Let us denote 𝑀 = sup N. Then for the number 𝑀 − 1∃𝑛 ∈ N : 𝑛 > 𝑀 − 1. But then 𝑛 + 1 > 𝑀 , i.e. 𝑀 is not sup N. Contradiction. Corollary 20. (from Archimedes’ principle) 1 ∀𝜀 > 0∃𝑛 ∈ N : < 𝜀. 𝑛 Corollary 21. 1 If 𝑥 > 0 and ∀𝑛 ∈ N𝑥 < 𝑛, then 𝑥 = 0. Corollary 22. For ∀𝑎, 𝑏 ∈ R : 𝑎 < 𝑏∃𝑟 ∈ Q : 𝑎 < 𝑟 < 𝑏. §4 Cantor’s principle. Examples of number sets: segment, interval, half-interval, ray, line, empty set, N, Q, etc. From sets, one can form systems of sets. For example, 1 {[0; ], 𝑛 ∈ N}, 𝑛 1 {(0; ), 𝑛 ∈ N}. 𝑛 Statement ¶ For the system of intervals {(0; 1/𝑛), 𝑛 ∈ N} there is no point common to all intervals, that is ∞ ⋂︁ 1 (0; ) = ∅. 𝑛=1 𝑛 Definition 23. Let there be a system of sets 𝑋1 , 𝑋2 ,... , 𝑋𝑛 ,.... If 𝑋1 ⊃ 𝑋2 ⊃ · · · ⊃ 𝑋𝑛 ⊃... , those. ∀𝑛 ∈ N𝑋𝑛+1 ⊂ 𝑋𝑛 , then this system is called a system of nested sets. Cantor’s principle. For any system of nested segments 𝐼1 ⊃ 𝐼2 ⊃ · · · ⊃ 𝐼𝑛 ⊃... there is a point common to all segments, that is ∞ ⋂︁ ∃𝑐 ∈ R : 𝑐 ∈ 𝐼𝑛. 𝑛=1 If, in addition, the system of segments is such that ∀𝜀 > 0 there is a segment whose length is less 𝜀, then the point 𝑐 is only one. Proof.¶ Let 𝐼 = [𝑎𝑛 ; 𝑏𝑛 ]. Consider two sets: 𝐴 = {𝑎𝑛 |𝑛 ∈ N} and 𝐵 = {N} (𝐴— left ends of 𝐼𝑛 , 𝐵— right ends of 𝑓 : 𝑋 → 𝑌 ). It is not difficult to prove (by contradiction) that 𝑦 = 𝑓 (𝑥) 𝑎𝑚 6 𝑏𝑛 (otherwise 𝑎𝑛 6 𝑏𝑛 < 𝑎𝑚 6 𝑏𝑚 , and they would not intersect). So, according to the axiom of completeness, ∃𝑐 ∈ R : 𝑎𝑚 6 𝑐 6 𝑏𝑛 , ∀𝑚, 𝑛 ∈ N, including for the case 𝑚 = 𝑛 too. So, ∞ ⋂︁ 𝑎𝑛 6 𝑐 6 𝑏𝑛 , ∀𝑛 ∈ N ⇒ 𝑐 ∈ [𝑎𝑛 ; 𝑏𝑛 ]. 𝑛=1 Let’s now say ∃𝑐1 ̸= 𝑐2 , common to all segments. Let 𝑐1 − 𝑐2 > 0. let’s take as 𝜀 = 𝑐1 − 𝑐2. Then there is a segment 𝐼𝑛 of length less 𝜀. Since ∀𝑛 ∈ N : 𝑎𝑛 6 𝑐1 6 𝑏𝑛 and 𝑎𝑛 6 𝑐2 6 𝑏𝑛 , then the length 𝐼𝑛 is equal to 𝑏𝑛 − 𝑎𝑛 > 𝑐1 − 𝑐2. But, by construction, the length of 𝐼𝑛 is less 𝑐1 − 𝑐2. Contradiction.