Page:Cantortransfinite.djvu/110

Rh B. If two aggregates $$M$$ and $$N$$ are such that $$M$$ is equivalent to a part $$N_1$$ of $$N$$ and $$N$$ to a part $$M_1$$ of $$M$$, then $$M$$ and $$N$$ are equivalent;

C. If $$M_1$$ is a part of an aggregate $$M$$, $$M_2$$ is a part of the aggregate $$M_1$$, and if the aggregates $$M$$ and $$M_2$$ are equivalent, then $$M_1$$ is equivalent to both $$M$$ and $$M_2$$;

D. If, with two aggregates $$M$$ and $$N$$, $$N$$ is equivalent neither to $$M$$ nor to a part of $$M$$, there is a part $$N_1$$ of $$N$$ that is equivalent to $$M$$;

E. If two aggregates $$M$$ and $$N$$ are not equivalent, and there is a part $$N_1$$ of $$N$$ that is equivalent to $$M$$, then no part of $$M$$ is equivalent to $$N$$.

[485] The Addition and Multiplication of Powers The union of two aggregates $$M$$ and $$N$$ which have no common elements was denoted in § 1, (2), by $$(M, N)$$. We call it the "union-aggregate (Vereinigungsmenge) of $$M$$ and $$N$$."

If $$M'$$ and $$N'$$ are two other aggregates without common elements, and if $$M\sim M'$$ and $$N\sim N'$$, we saw that we have $(M,N)\sim (M', N')$. Hence the cardinal number of $$(M, N)$$ only depends upon the cardinal numbers $$\overline\overline{M} = \mathfrak{a}$$ and $$\overline\overline{N} = \mathfrak{b}$$.

This leads to the definition of the sum of a and b. We put (1)