By Helena Rasiowa (Auth.)

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In this case the indexed family (An)nejr is a sequence of subsets of the set of real numbers. The first four terms of that sequence are written out below: Ai = {x e ®\ -1 Α2 = {xe 1 < χ < 1+y}, ίζ χ ίζ 2}, 1. CONCEPT OF GENERALIZED UNION AND INTERSECTION A3 = {xe ^ 0t\ = {xel: (Ill) Let Τ = 0t (0t—the -1 51 < x < 1< χ < set of real numbers) and let 91 be the family of all subsets of 0t. Let m\ f(t) = At = {xe 2 \x\

Prove that Take also d take Bn = Axv = Λ and C„ = Λ π —C4„u ... vAn for every for /i = 2, 3, ...

10. Prove that, for any non-empty family SR of subsets of a universe X, there exists a least field & of subsets of X which contains SR, that is, which is contained in any field of subsets of X containing SR. , sets of the form (AnnA12n ... nAlni)v ... v(AmlnAm2n ... nAm„ ), where A-^ e SR or A\j == —Bij and B,j e SR, is the least field that contains SR. CHAPTER II NATURAL NUMBERS. PROOFS BY INDUCTION 1. An axiomatic approach to natural numbers. The principle of induction The numbers 1, 2 , 3 , ...