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Additional resources for Algebra Vol. I

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We extend this definition to the empty set, giving it the cardinal number zero (= 0). A corresponding concept arises for infinite sets, if we assign some object to every infinite set, which we then call the cardinal number of this infinite set so that the same cardinal number should correspond to two sets if and only if they are equipotent. THEOREM 10. Every infinite subset of a countably infinite set is countable. >) is countable. ) in the following manner. Letf(l) denote, on account of Theorem 5, the least element of S.

We now prove that as a consequence of associativity, one element can have at most one inverse. 3) that fi=NE=NIcY)=(a)Y=ey=Y. The inverse of a will be denoted henceforth by a-1. 4) is valid. For a one-to-one mapping a we have formerly denoted the inverse mapping by a-1. Both notations are in accordance with each other, since (r-la = aa-1 = 1 is the identical mapping. 5) are (in S) solvable at all, and in case of solvability how many solutions they have. 5) are soluble for each pair a, P, then we call the multiplication invertible.

4-<... of the index set. CHAPTER 11 STRUCTURES § 17. Compositions In the set of the natural numbers certain "compositions" are defined, for example the sum a + b and the product ab of two numbers a, b. A further important example is the product a,B of two mappings a, ,B of a set into itself, as defined in § 4. We shall generalize these and similar well-known examples below. >, if an element of S is uniquely assigned to each ordered pair a, j9 of elements of S in a certain way. We denote this element by a 0 j3 where 0 is the sign of composition.