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By Redei L.

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Equipotent Sets CANTOR calls two sets A, B equivalent or equipotent, if there exists a one-to- one mapping of A onto B. We then write : A B. In this case we also say that A and B are of equal potency; otherwise, they are of different potency. : is indeed an equivalence relation. Since the set A is mapped one-to-one onto itself by its identical mapping, then A A holds. , there exists a one-to-one mapping a of A onto B, then the inverse mapping a-1 is a one-to-one mapping of B onto A whence B x A.

7) . , x,,,+i) Evidently k(n) lies in S. e. ). 7), k(n + 1) = (k(n), f(n + 1). ). 8), we find that f is the required function. Consequently Theorem 7 is proved. The above-mentioned axioms I, II agree with the axioms of Peano, by which he was the first to found the concept of the natural numbers axiomatically. These axioms are 1. 1 is a natural number. 2. To every natural number n there belongs uniquely a natural number n'. 3. For every n, n' 0 1. 4. From m' = n' it follows that m = n. 5. ,07 contains the element I and with every element n also n'.

Not lying in the same plane), then the set of all alas + alas + alas exhausts, as is well known, all the vectors of the space considered, a circumstance which is evidently to be attributed to the use of the set of the real numbers as the operator domain. Mostly two cases occur; either the sets O, S are disjoint, or 0 c S. This second case arises when we define a multiplication (and possibly further compositions) in S and we identify the operator product as (a E O c S, a E S) with the product existing in S.

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Algebra Vol. I by Redei L.

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