Page:Cantortransfinite.djvu/114

Rh $f(n_0)=m_0$ $f(n)=m_1$ for all $$n$$'s which are different from $$n_0$$.

The totality of different coverings of N with M forms a definite aggregate with the elements $$f(N)$$; we call it the "covering-aggregate (Belegungsmenge) of $$N$$ with $$M$$" and denote it by $$(N|M)$$. Thus: (2) If $$M\sim M'$$ and $$N\sim N'$$, we easily find that (3) Thus the cardinal number of $$(N|M)$$ depends only on the cardinal numbers $$\overline\overline{M} = \mathfrak{a}$$ and $$\overline\overline{N} = \mathfrak{b}$$; it serves us for the definition of $$\mathfrak{a}^\mathfrak{b}$$ : (4) For any three aggregates, $$M, N, P$$, we easily prove the theorems: (5) (6) (7) from which, if we put $$\overline\overline{P} = \mathfrak{c}$$, we have, by (4) and by paying attention to § 3, the theorems for any three cardinal numbers, $$\mathfrak{a}$$, $$\mathfrak{b}$$, and $$\mathfrak{c}$$: (8) (9) (10)