# Algebra Vol. I by Redei L. By Redei L.

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A Treatise on Universal Algebra with Applications

Symbolic Reasoning allied to dull A lgebra. the executive examples of such platforms are Hamilton sQ uaternions, Grassmann sC alculus of Extension and Boole sS ymbolic common sense. Such algebras have an intrinsic worth for separate distinct learn; they are also priceless of a comparative learn, for the sake of the sunshine thereby thrown at the normal concept of symbolic reasoning, and on algebraic symbolism particularly.

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Also BIRKHOFF (1948). § 15. Theorem of Well-ordering THEOREM 19 (theorem Of ZERMELO or the theorem of well-ordering). Any set can be well-ordered. Before the proof we note by way of introduction that: Given two semiordered sets A, B, we shall only then say that A is a subset of B (or B is an o verset of A), if, first of all, the set B contains the set A and any relation x < y (x, y E A) which is true in B, is also true in A. ) Naturally this also applies to the concept of the cuts of a semiordered set.

Consequently the assertion is proved. , n) are called finite, any other set is called infinite. , an) . THEOREM 8. A set is finite if, and only if, it is not equivalent to any of its proper subsets. First, we prove that it is impossible to map a finite set one-to-one onto any proper subset of itself. , n> . In the case n = l the statement is true. We assume its truth for any n and prove that it is true for n + 1. , that a one-to-one mapping x -+ ox of the set <1, . , n + 1> onto a proper subset S of it exists.

1), onto a proper subset of itself. This contradicts the induction assumption. 1) does not hold, then n + 1 E S. Then there certainly is an a (= 1, . , n) not belonging to S. 2) holds. 2)]. , n). 1), whereby we have proved the above statement. Later we shall prove the second part of Theorem 8. > by the one-to-one mapping (123... I` 234... > of all the natural numbers cannot be finite. Consequently it is infinite. ) of the natural numbers, we call countably infinite. The finite and the countably infinite sets are called countable sets.