ELEMENTS OF REAL ANALYSIS PDF
The Elements of Real Analysis. Robert G. Bartle. Department of Mothematics, University of Illinois. John Wiley & Sons, Inc., New York • London • Sydney. PDF | On Jul 1, , Akhilesh Yadav and others published Elements of Real Analysis. Basic Elements of Real Analysis provides, in a brief and elementary way, the most book A First Course in Real Analysis or the text Principles of Mathematical.
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Bartle, Robert - The Elements of Real Analysis - Ebook download as PDF File . pdf), Text File .txt) or read book online. The Elements of Real Analysis Robert G. Bartle Department of Mothematics, University of Illinois. John Wiley & Sons, Inc., New York • London • Sydney. This is a text for a two-term course in introductory real analysis for junior or senior math- . two elements, which we denote by 0 and 1, with addition defined by.
This last visualization makes clear the distinction between. Certainly it is useful to distinguish between a machine and its outputs. In doing SO, some tare needs to be exercised concerning the domain of the resulting function. Let j be a function with domain 9 f in A and range a j in B and let g be a function with domain ZD g in B and range a g in C. The composition g 0 j note the order!
The fact that g 0j is actually a function and not an arbitrary subset of A X C is a consequence of the fact that j and g are functions see Exercise 2. Figure 2. Composition of functions. Note that the formula expressing g 0 h has meaning for values of x other than those in the domain of g 0 h.
We claim that if f is a one-one function from A to B, then the set of ordered pairs in B X A obtained by interchanging the first and second members of ordered pairs in f yields a function g which is also one-one. The connections between f and g are: w! Let f be a one-one function with domain 9 f in Aandrange6l f inB. The function g is called the function inverse to f and we ordinarily denote g by f-l. It is readily seen that F is not one-one; in fact, the ordered pairs 2,4 , -2, 4 both belong to F.
Since F is not one-one, it does not have an inverse. The function gl inverse to f is called the negative square root faction and is denoted by! The inverse function cari be interpreted from the mapping point of view. The inverse function. The inverse function j-l maps the element b into this unique element a.
Direct and Inverse Images Once again, let f be an arbitrary function with domain a f in A and range a f in B. We do not assumethat j is one-one. For the sake of brevity, we sometimes denote the direct image of a set E under f by the notation f E. If E consists of the single point p in 3 f , then the set f E consists of the single point f p.
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Certain properties of sets are preserved under the direct image, as we now show. Therefore, we conclude that f E n F c f E nf F. Direct images. It Will be seen in Exercise 2. J that it is not possible to replace the inclusion sign in b by equality, in general.
We now introduce the notion of the inverse image of a set under a function. Note that it is not required that the function be one-one. Once again, we emphasize that f need not be one-one SO that the inverse function f-l need not exist.
It Will probably corne as a surprise to the reader to learn that the inverse image is better behaved than the direct image. This is shown in the next result. Exercises 2.
Prove that Dehnition 2. Describethis set in words. Supposef is a one-onefunction. Showthat f is one-oneand that 63 f 5 D g. Hence f E n F is a proper subsetof f E nf F.
If f, E, F are asin Exercise2. Showthat if f is a one-onemappingof X f into R f and if H is a subset of 6l f , then the inverse imageof H under f coincideswith the direct imageof H under the inverse function f-l.
If f and g are as in Definition 2.
Same with B and b, and C and c. Also, when we say an element a is in a set A, we use the symbol to show it. And if something is not in a set use. We can see that 1 A, but 5 A Equality Two sets are equal if they have precisely the same members.
Now, at first glance they may not seem equal, so we may have to examine them closely! They both contain 1. They both contain 2.
And 3, And 4. And we have checked every element of both sets, so: Yes, they are equal! They both contain exactly the members 1, 2 and 3.
It doesn't matter where each member appears, so long as it is there. Subsets When we define a set, if we take pieces of that set, we can form what is called a subset.
In general: A is a subset of B if and only if every element of A is in B. So let's use this definition in some examples.
So far so good. That's all the elements of A, and every single one is in B, so we're done. But remember, that doesn't matter, we only look at the elements in A. Let's try a harder example. Example: Let A be all multiples of 4 and B be all multiples of 2.
Is A a subset of B? And is B a subset of A? Well, we can't check every element in these sets, because they have an infinite number of elements.
So we need to get an idea of what the elements look like in each, and then compare them. Let A be a set. Is every element in A an element in A? Yes, I wrote that correctly. Well, umm, yes of course, right?
So doesn't that mean that A is a subset of A? This doesn't seem very proper, does it? We want our subsets to be proper.
Bartle, Robert - The Elements of Real Analysis
So we introduce what else but proper subsets. A is a proper subset of B if and only if every element in A is also in B, and there exists at least one element in B that is not in A. This little piece at the end is only there to make sure that A is not a proper subset of itself. Otherwise, a proper subset is exactly the same as a normal subset.
Notice that if A is a proper subset of B, then it is also a subset of B. Or we can say that A is not a subset of B by A B "A is not a subset of B" When we talk about proper subsets, we take out the line underneath and so it becomes A B or if we want to say the opposite, A B.
This is probably the weirdest thing about sets. As an example, think of the set of piano keys on a guitar. It is a set with no elements. This is known as the Empty Set or Null Set. There aren't any elements in it. Not one. So what's so weird about the empty set? Well, that part comes next. Empty Set and Subsets So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. Is the empty set a subset of A? Going back to our definition of subsets, if every element in the empty set is also in A, then the empty set is a subset of A.
But what if we have no elements? It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.
A good way to think about it is: we can't find any elements in the empty set that aren't in A, so it must be that all elements in the empty set are in A. So the answer to the posed question is a resounding yes.In order to show that 8 has an infimum. The reader should realize that it is more difficult to show that a set is.
BunyakovskiiS and H.
Thus the domain of the absolute value function is a11of F. In formulating the next definition.