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1.1) that¬(P ∧ Q) ⇔ ¬P ∨ ¬Q,¬(P ∨ Q) ⇔ ¬P ∧ ¬Q,¬(P ⇒ Q) ⇔ P ∧ ¬Q.On the basis of what has just been said, one can conclude, for example, that¬ (∀x > a)P ⇔ (∃x > a)¬P .It would of course be wrong to express the right-hand side of this last relation as(∃x ≤ a)¬P .Indeed, ¬ (∀x > a)P := ¬ ∀x x ∈ R ∧ x > a ⇒ P (x) ⇔ ⇔ ∃x ¬ x ∈ R ∧ x > a ⇒ P (x) ⇔⇔ ∃x (x ∈ R ∧ x > a) ∧ ¬P (x) =: (∃x > a)¬P .If we take into account the structure of an arbitrary statement mentioned above,we can now use the negations just constructed for the simplest statements to formthe negation of any particular statement.For example,¬ lim f (x) = A ⇔ ∃ε > 0 ∀δ > 0 ∃x ∈ Rx→a0 < |x − a| < δ ∧ f (x) − A ≥ ε .The practical importance of the rule for forming a negation is connected, in particular, with the method of proof by contradiction, in which the truth of a statementP is deduced from the fact that the statement ¬P is false.321 Some General Mathematical Concepts and Notation1.4.4 Exercises1.
a) Prove the equipotence of the closed interval {x ∈ R | 0 ≤ x ≤ 1} and the openinterval {x ∈ R | 0 < x < 1} of the real line R both using the Schröder–Bernsteintheorem and by direct exhibition of a suitable bijection.b) Analyze the following proof of the Schröder–Bernstein theorem:(card X ≤ card Y ) ∧ (card Y ≤ card X) ⇒ (card X = card Y ).Proof It suffices to prove that if the sets X, Y , and Z are such that X ⊃ Y ⊃ Z andcard X = card Z, then card X = card Y .
Let f : X → Z be a bijection. A bijectiong : X → Y can be defined, for example, as follows:g(x) =f (x),xif x ∈ f n (X)\f n (Y ) for some n ∈ N,otherwise.Here f n = f ◦ · · · ◦ f is the nth iteration of the mapping f and N is the set of naturalnumbers.2. a) Starting from the definition of a pair, verify that the definition of the directproduct X × Y of sets X and Y given in Sect. 1.4.2 is unambiguous, that is, the setP(P(X) ∪ P(Y )) contains all ordered pairs (x, y) in which x ∈ X and y ∈ Y .b) Show that the mappings f : X → Y from one given set X into another givenset Y themselves form a set M(X, Y ).c) Verify that if R is a set of ordered pairs (that is, a relation), then the firstelements of the pairs belonging to R (like the second elements) form a set.3.
a) Using the axioms of extensionality, pairing, separation, union, and infinity,verify that the following statements hold for the elements of the set N0 of naturalnumbers in the sense of von Neumann:10203040x = y ⇒ x+ = y+;(∀x ∈ N0 ) (x + = ∅);x + = y + ⇒ x = y;(∀x ∈ N0 ) (x = ∅ ⇒ (∃y ∈ N0 ) (x = y + )).b) Using the fact that N0 is an inductive set, show that the following statementshold for any of its elements x and y (which in turn are themselves sets):1020304050card x ≤ card x + ;card ∅ < card x + ;card x < card y ⇔ card x + < card y + ;card x < card x + ;card x < card y ⇒ card x + ≤ card y;1.4 Supplementary Material3360 x = y ⇔ card x = card y;70 (x ⊂ y) ∨ (x ⊃ y).c) Show that in any subset X of N0 there exists a (minimal) element xm suchthat (∀x ∈ X) (card xm ≤ card x).
(If you have difficulty doing so, come back to thisproblem after reading Chap. 2.)4. We shall deal only with sets. Since a set consisting of different elements mayitself be an element of another set, logicians usually denote all sets by uppercaseletters. In the present exercise, it is very convenient to do so.a) Verify that the statement∀x ∃y ∀z z ∈ y ⇔ ∃w (z ∈ w ∧ w ∈ x)expresses the axiom of union, according to which y is the union of the sets belongingto x.b) State which axioms of set theory are represented by the following statements:∀x ∀y ∀z (z ∈ x ⇔ z ∈ y) ⇔ x = y ,∀x ∀y ∃z ∀v v ∈ z ⇔ (v = x ∨ v = y) ,∀x ∃y ∀z z ∈ y ⇔ ∀u (u ∈ z ⇒ u ∈ x) , ∃x ∀y ¬∃z (z ∈ y) ⇒ y ∈ x ∧ ∧ ∀w w ∈ x ⇒ ∀u ∀v v ∈ u ⇔ (v = w ∨ v ∈ w) ⇒ u ∈ x .c) Verify that the formula∀z z ∈ f ⇒ ∃x1 ∃y1 x1 ∈ x ∧ y1 ∈ y ∧ z = (x1 , y1 ) ∧∧ ∀x1 x1 ∈ x ⇒ ∃y1 ∃z y1 ∈ y ∧ z = (x1 , y1 ) ∧ z ∈ f ∧∧ ∀x1 ∀y1 ∀y2 ∃z1 ∃z2 z1 ∈ f ∧ z2 ∈ f ∧ z1 = (x1 , y1 ) ∧ z2 = (x1 , y2 ) ⇒⇒ y1 = y 2imposes three successive restrictions on the set f : f is a subset of x × y; the projection of f on x is equal to x; to each element x1 of x there corresponds exactlyone y1 in y such that (x1 , y1 ) ∈ f .Thus what we have here is a definition of a mapping f : x → y.This example shows yet again that the formal expression of a statement is by nomeans always the shortest and most transparent in comparison with its expressionin ordinary language.
Taking this circumstance into account, we shall henceforthuse logical symbolism only to the extent that it seems useful to us to achieve greatercompactness or clarity of exposition.5. Let f : X → Y be a mapping. Write the logical negation of each of the followingstatements:341 Some General Mathematical Concepts and Notationa) f is surjective;b) f is injective;c) f is bijective.6. Let X and Y be sets and f ⊂ X × Y . Write what it means to say that the set f isnot a function.Chapter 2The Real NumbersMathematical theories, as a rule, find uses because they make it possible to transform one set of numbers (the initial data) into another set of numbers constitutingthe intermediate or final purpose of the computations.
For that reason numericalvalued functions occupy a special place in mathematics and its applications. Thesefunctions (more precisely, the so-called differentiable functions) constitute the mainobject of study of classical analysis. But, as you may already have sensed from yourschool experience, and as will soon be confirmed, any description of the propertiesof these functions that is at all complete from the point of view of modern mathematics is impossible without a precise definition of the set of real numbers, on whichthese functions operate.Numbers in mathematics are like time in physics: everyone knows what they are,and only experts find them hard to understand.
This is one of the basic mathematical abstractions, which seems destined to undergo significant further development.A very full separate course could be devoted to this subject. At present we intendonly to unify what is basically already known to the reader about real numbers fromhigh school, exhibiting as axioms the fundamental and independent properties ofnumbers. In doing this, our purpose is to give a precise definition of real numberssuitable for subsequent mathematical use, paying particular attention to their property of completeness or continuity, which contains the germ of the idea of passageto the limit – the basic nonarithmetical operation of analysis.2.1 The Axiom System and Some General Properties of the Setof Real Numbers2.1.1 Definition of the Set of Real NumbersDefinition 1 A set R is called the set of real numbers and its elements are realnumbers if the following list of conditions holds, called the axiom system of thereal numbers.© Springer-Verlag Berlin Heidelberg 2015V.A.
Zorich, Mathematical Analysis I, Universitext,DOI 10.1007/978-3-662-48792-1_235362The Real Numbers(I) (A XIOMS FOR ADDITION) An operation+ : R × R → R,(the operation of addition) is defined, assigning to each ordered pair (x, y) of elements x, y of R a certain element x + y ∈ R, called the sum of x and y.
Thisoperation satisfies the following conditions:1+ . There exists a neutral, or identity element 0 (called zero) such thatx +0=0+x =xfor every x ∈ R.2+ . For every element x ∈ R there exists an element −x ∈ R called the negative ofx such thatx + (−x) = (−x) + x = 0.3+ . The operation + is associative, that is, the relationx + (y + z) = (x + y) + zholds for any elements x, y, z of R.4+ . The operation + is commutative, that is,x+y =y+xfor any elements x, y of R.If an operation is defined on a set G satisfying Axioms 1+ , 2+ , and 3+ , we saythat a group structure is defined on G or that G is a group. If the operation is calledaddition, the group is called an additive group.
If it is also known that the operation is commutative, that is, condition 4+ holds, the group is called commutative orAbelian.1Thus, Axioms 1+ –4+ assert that R is an additive abelian group.(II) (A XIOMS FOR MULTIPLICATION) An operation• : R × R → R,(the operation of multiplication) is defined, assigning to each ordered pair (x, y) ofelements x, y of R a certain element x · y ∈ R, called the product of x and y.
Thisoperation satisfies the following conditions:1 N.H. Abel (1802–1829) – outstanding Norwegian mathematician, who proved that the generalalgebraic equation of degree higher than four cannot be solved by radicals.2.1 Axioms and Properties of Real Numbers371• . There exists a neutral, or identity element 1 ∈ R\0 (called one) such thatx ·1=1·x =xfor every x ∈ R.2• . For every element x ∈ R\0 there exists an element x −1 ∈ R, called the inverseor reciprocal of x, such thatx · x −1 = x −1 · x = 1.3• . The operation • is associative, that is, the relationx · (y · z) = (x · y) · zholds for any elements x, y, z of R.4• . The operation • is commutative, that is,x ·y =y ·xfor any elements x, y of R.We remark that with respect to the operation of multiplication the set R\0, as onecan verify, is a (multiplicative) group.(I, II) (T HE CONNECTION BETWEEN ADDITION AND MULTIPLICATION) Multiplication is distributive with respect to addition, that is(x + y)z = xz + yzfor all x, y, z ∈ R.We remark that by the commutativity of multiplication, this equality continues tohold if the order of the factors is reversed on either side.If two operations satisfying these axioms are defined on a set G, then G is calleda field.(III) (O RDER AXIOMS) Between elements of R there is a relation ≤, that is, forelements x, y ∈ R one can determine whether x ≤ y or not.
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