By Hilborn R C

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If S has a minimum element, we denote it by min S. A number L is called a greatest lower bound (or injîmum) of S if (a) L is a lower bound for S, and (b) no number greater than L is a lower bound for S. The infimum of S, when it exists, is uniquely determined and we denote it by inf S. If S has a minimum element, then min S = inf S. Using Axiom 10, we cari prove the following. 27. Every nonempty set S that is bounded below has a greatest lower bound; that is, there is a real number L such that L = inf S.

4. -(-a) = a. 5. a(b - c) = ab ‘- ac. 6. 0 *a = a * 0 = 0. 7. C A N C E L L A T I O N L A W F O R M U L T I P L I C A T I O N . Zf ab = ac and a # 0, then b = c. 8. POSSIBILITY OF DIVISION. Given a and b with a # 0, there is exactly one x such that ax = b. This x is denoted by bja or g and is called the quotient of b and a. particular, lia is also written aa1 and is called the reciprocal of a. 9. If a # 0, then b/a = b * a-l. 10. 11. Zfab=O,thena=Oorb=O. 12. (-a)b = -(ah) and (-a)(-b) = ab. 13.

Capital script letters d, g, %‘, . . are used to denote classes. The usual terminology and notation of set theory applies, of course, to classes. Z means that every set in I is also in 9, and SO forth. Exercises 15 defined as the set of those elements which belong to at least one of the sets in 9 and is denoted by the symbol UA. AET If 9 is a finite collection of sets, say 9 = {A, , A,, . . , A,}, we Write *;-&A =Cl&= AI u A, u . . u A, . Similarly, the intersection of a11 the sets in 9 is defined to be the set of those elements which belong to every one of the sets in 9; it is denoted by the symbol ALLA.