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

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

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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Additional resources for Analysis (2 volume set) (Texts and Readings in Mathematics)

Example text

An; this is known as "finite choice". 4. 8. Note that the empty set is not the same thing as the natural number 0. One is a set; the other is a number. 6. 2 was the only axio~ that set theory had, then set theory could be quite boring, as there might be just a single set in existence, the empty set. We now present further axioms to enrich the class of sets available. 3 (Singleton sets and pair sets). , for every object y, we have yE {a} if and only if y =a; we refer to {a} as the singleton set whose element is a.

2 again, 2++ = 3 is a natural number. D It may· seem that this is enough to describe the natural num- bers. 5. Consider a number system which consists of the numbers 0, 1, 2, 3, in which the increment operation wraps back from 3 to 0. More precisely 0++ is equal to 1, 1++ is equal to 2, 2++ is equal to 3, but 3++ is equal to 0 (and also equal to 4, by definition of 4). This type of thing actually happens in real life, when one uses a computer to try to store a natural number: if one starts at 0 and performs the increment operation repeatedly, eventually the computer will overflow its memory and the number will wrap around back to 0 (though this may take quite a large number of incrementation operations, for instance a two-byte representation of an integer will wrap around only after 65,536 increments).

AUB)UC = AU (BUG)). Also, we have A U A = A U 0 = 0 U A = A. 1. 3. 4, we need to show that every element x of (AUB)U C is an element of A U (B U C), and vice versa. So suppose first that xis an element of (A U B) U C. 4, this means that at least one of x E A U B or x E C is true. We now divide into two cases. 4 again we have x EAU (B U C). 4 again x E A or x E B. 4 we have x E B U C and hence x E A U (B U C). Thus in all cases we see that every element of (A U B) U C lies in A U (B U C). A similar argument shows that every element of A U (B U C) lies in (A U B) U C, and so (A U B) U C =A U (B U C) as desired.

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


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