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By Terence Tao

ISBN-10: 8185931623

ISBN-13: 9788185931623

ISBN-10: 8185931631

ISBN-13: 9788185931630

ISBN-10: 818593195X

ISBN-13: 9788185931951

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6 is not equal to 2. Proof. Suppose for sake of contradiction that 6 = 2. Then 5++ = = 1, so that 4++ = 0++. 4 again we then have 4 = 0, which contradicts our previous proposition. 4 we have 5 As one can see from this proposition, it now looks like we can keep all of the natural numbers distinct from each other. 2) allow us to confirm that 0, 1, 2, 3, ... 9. 5, ... }. 4 are still satisfied for this set. 5. But it is difficult to quantify what we mean by "can be obtained from" without already using the natural numbers, which we are trying to define.

10. ). Similarly, the singleton set {{0}} and the pair set {0, {0}} are also sets. 2). As the above examples show, we can now create quite a few sets; however, the sets we make are still fairly small (each set that we can build consists of no more than two elements, so far). The next axiom allows us to build somewhat larger sets than before. 4 (Pairwise union). Given any two sets A, B, there exists a set A U B, called the union A U B of A and B, whose elements consists of all the elements which belong to A or B or both.

This will teach you a new skill - how to prove complicated properties from simpler ones. You will find that even though a statement may be "obvious", it may not be easy to prove; the material here will give you plenty of practice in doing so, and in the process will lead you to think about why an obvious statement really is obvious. One skill in particular that you will pick up here is the use of mathematical induction, which is a basic tool in proving things in many areas of mathematics. So in the first few chapters we will re-acquaint you with various number systems that are used in real analysis.

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Analysis (2 volume set) (Texts and Readings in Mathematics) by Terence Tao

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