apuntes análisis real

7/30/2019 Apuntes Anlisis Real

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Review of Real Analysis

Alx Ruiz

March 23, 2013

NAIVE SET THEORY by PAUL R. HALMOS

Axiom of Pairing. For any two sets there exists a set that they both belongto.

Thus, for instance, the axiom of pairing, in unabbreviated form, saysthat ifa and b are sets, then there exists a set A such that a A and b A.One consequence (in fact an equivalent formulation) of the axiom of pairingis that for any two sets there exists a set that contains both of them andnothing else. Indeed, ifa and b are sets, and ifA is a set such that a A andb A, then we can apply the axiom of specification to A with the sentencex=a or x=b. The result is the set

{x A : x = a x = b}

and that set, clearly, contains just a and b. The axiom of extension impliesthat there can be only one set with this property. The usual symbol for thatset is

{a, b};

the set is called the pair (or, by way of emphatic comparison with a subse-quent concept, the unordered pair) formed by a and b.

If, temporarily, we refer to the sentence x=a or x=b as S(x), we mayexpress the axiom of pairing by saying that there exists a set B such that

x B if and only ifS(x). (1)

The axiom of specification, applied to a set A, asserts the existence of aset B such that

x B if and only if (x A S(x)) (2)

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All the remaining principles of set construction are pseudo-special cases

of the axiom of specification in the sense in which (1) is a pseudo-specialcase of (2). They all assert the existence of a set specified by a certain con-dition; if it were known in advance that there exists a set containing all thespecified elements, then the existence of a set containing just them wouldindeed follow as a special case of the axiom of specification.

If a is a set, we may form the unordered pair {a, a}. That unorderedpair is denoted by

{a}

is called the singletonofa; it is uniquely characterized by the statement thatit has a as its only element. Thus, for instance, and {} are very different

sets; the former has no elements, whereas the latter has the unique element.To say that a A is equivalent to saying that {a} A.

The axiom of pairing ensures that every set is an element of some setand that any two sets are simultaneously elements of some one and the sameset. Another pertinent comment is that from the assumptions we have madeso far we can infer the existence of very many sets indeed. For examplesconsider the sets , {}, {{}}, {{{}}}, etc.; consider the pairs, such as{, {}}, formed by any two of them; consider the pairs formed by any twosuch pairs, or else the mixed pairs formed by any singleton and any pair;

and proceed so on ad infinitum.

If A is a set and S(x) is an arbitrary sentence, it is permissible to form{x : x A and S(x)}; the set is the same as {x A : S(x)}. As furtherexamples, we note that

{x: x = x} =

and{x: x = a} = {a}.

Axiom of powers. For each set there exists a collection of sets that con-

tains among its elements all the subsets of the given set.

In other words, ifE is a set, then there exists a (collection) P such that ifX E, then X P. The set P described above may be larger than wanted;it may contain elements other than the subset of E. This is easy to remedy;

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just apply the axiom of specification to form the set {X P : X E}.

(Recall that X E says the same thing as for all x (if x X then x E).) Since, for every X, a necessary and sufficient condition that X belongto this set is that X be a subset of E, it follows that if we change notationand call this set P again, then

P = {X: X E}.

The set P is called the power set of E; the axiom of extension guarantees itsuniqueness. The dependence of P on E is denoted by writing P(E) insteadof just P.Because the set P(E) is very big in comparison with E, it is not easy to

give examples. If E = , the situation is clear enough; the set P() isthe singleton {}. The power set of singletons and pairs are also easilydescribable; we have

P({a}) = {, {a}}

andP({a, b}) = {, {a}, {b}, {a, b}}.

The power set of a triple has eight elements. The reader can probablyguess (and is hereby challenged to prove) the generalization that includesall these statements: the power set of a finite set with, say, n elements has 2n

elements. (Of course concepts like finite and 2n have no official standing

for us yet; this should not prevent them from being unofficially understood.)The occurrence of n as an exponent (the n-th power of 2) has something todo with the reason why a power set bears its name.IfC is a collection of subsets of a set E (that is, C is a subcollection ofP(E)),then write

D = {X P(E): X C }.

To be certain that the condition used in the definition of D is a sentence inthe precise technical sense, it must be rewritten in something like the form

f or some Y [Y C and for all x (x X if and only if (x E and x Y))].

The ordered pairofa and b, with first coordinate aand second coordinateb, is the set (a, b) defined by

(a, b) = {{a}, {a, b}}.

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Cartesin product of A and B; it is characterized by the fact that

A B = {x: x = (a, b) for some a in A and for some b in B.}

ANALISYS: AN INTRODUCTION TO PROOF by STEVEN R. LAY

Relation. Let A and B be sets. A relation between A and B is anysubset R of A B. We say that a A and b B are related by R if (a, b) R, aRb.Equivalence Relations. There are certain relations that are singled out be-

cause they possess the properties naturally associated with the idea of equal-ity. They are called equivalence relations, as we see in the following defini-tion.Definition. A relation R on a set S is an equivalence relationif it has thefollowing properties for all for all x, y, z in S:

1. Reflexive Property. xRx

2. Symmetric Property. If xRy, then yRx

3. Transitive Property. If xRy and yRz, then xRz

Function. Let A and B be sets. A function between A and B is a

nonempty relation f A B such that if (a, b) fand (a, b) f, thenb = b.

Domain. The domain offis the set of all first elements of members off.

domain f= {a: b B (a, b) f}

Range. The range offis the set of all second elements of members off.

range f= {b: a A (a, b) f}

The set B is referred to as the codomain of f.

If the domain off= all A, we say fis a functionfrom A into B and we writef: A B.

If (x, y) is a member of f, we often say that fmaps x onto y or that y

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is the image of x under f.

A function f: A B is called surjective(or to map A onto B) if B =range f.

A function f: A B is called injective (or one-to-one) if for all a anda in A, f(a) = f(a) implies that a = a.

Bijective if it is surjective and injective.

Suppose that f: A B. If C A, we let f(C) is called the image ofC in B. If D B, we let f1(D) represent the subset {x A: f(x) D}ofA. The set f1(D) is called the pre-imageofD in A (or f inverse ofD).

Composition of Functions.

If f and g are functions with f: A B and g : B C, then for anya A, f(a) B. But B is the domain of g, so g can be applied to f(a).This yields g(f(a)), an element of C. This correspondence is called thecomposition function of f and g and is denoted by gf. Thus

(gf)(a) = g(f(a)).

in terms of ordered pairs we have

gf = {(a, c) A C: b B (a, b) f and (b, c) g}.

Inverse Functions.

Let f: A B be bijective. The inverse function of fis the function f1

given by

f1 = {(y, x) B A: (x, y) f}.

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