By Terence Tao

ISBN-10: 8185931623

ISBN-13: 9788185931623

**Read Online or Download Analysis (2 volume set) (Texts and Readings in Mathematics) PDF**

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3. 6 (Multiplication preserves order). If a, b are natuml numbers such that a < b, and c is positive, then ac < be. 3. Multiplication 35 Proof. Since a < b, we have b = a+ d for some positive d. Multiplying by c and using the distributive law we obtain be = ae +de. Since d is positive, and c is positive, de is positive, and hence ac < be as desired. 3. 7 (Cancellation law). Let a, b, e be natuml numbers such that ac =be and e is non-zero. Then a= b. 8. 6 will allow for a "vir-· tual subtraction" which will eventually let us define genuine subtraction, this corollary provides a "virtual division" which will be needed to define genuine division later on.

4. Thus the "is an element of" relation E obeys the axiom of substitution (see Section 40 3. 7). Because of this, any new operation we define on sets will also obey the axiom of substitution, as long as we can define that operation purely in terms of the relation E. This is for instance the case for the remaining definitions in this section. ) Next, we turn to the issue of exactly which objects are sets and which objects are not. The situation is analogous to how we defined the natural numbers in the previous chapter; we started with a single natural number, 0, and started building more numbers out of 0 using the increment operation.

Addition 27 Note how all of the axioms had to be used here. In a system which had some sort of wrap-around, recursive definitions would not work because some elements of the sequence would constantly be redefined. 5, in which 3-t+ = 0, then there would be (at least) two conflicting definitions for ao, either cor Ja(a3)). s would never be defined. Recursive definitions are very powerful; for instance, we can use them to define addition and multiplication, to which we now turn. 2 Addition The natural number system is very bare right now: we have only one operation - increment - and a handful of axioms.