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B. If two aggregates ${\displaystyle M}$ and ${\displaystyle N}$ are such that ${\displaystyle M}$ is equivalent to a part ${\displaystyle N_{1}}$ of ${\displaystyle N}$ and ${\displaystyle N}$ to a part ${\displaystyle M_{1}}$ of ${\displaystyle M}$, then ${\displaystyle M}$ and ${\displaystyle N}$ are equivalent;

C. If ${\displaystyle M_{1}}$ is a part of an aggregate ${\displaystyle M}$, ${\displaystyle M_{2}}$ is a part of the aggregate ${\displaystyle M_{1}}$, and if the aggregates ${\displaystyle M}$ and ${\displaystyle M_{2}}$ are equivalent, then ${\displaystyle M_{1}}$ is equivalent to both ${\displaystyle M}$ and ${\displaystyle M_{2}}$;

D. If, with two aggregates ${\displaystyle M}$ and ${\displaystyle N}$, ${\displaystyle N}$ is equivalent neither to ${\displaystyle M}$ nor to a part of ${\displaystyle M}$, there is a part ${\displaystyle N_{1}}$ of ${\displaystyle N}$ that is equivalent to ${\displaystyle M}$;

E. If two aggregates ${\displaystyle M}$ and ${\displaystyle N}$ are not equivalent, and there is a part ${\displaystyle N_{1}}$ of ${\displaystyle N}$ that is equivalent to ${\displaystyle M}$, then no part of ${\displaystyle M}$ is equivalent to ${\displaystyle N}$.

The union of two aggregates ${\displaystyle M}$ and ${\displaystyle N}$ which have no common elements was denoted in § 1, (2), by ${\displaystyle (M,N)}$. We call it the "union-aggregate (Vereinigungsmenge) of ${\displaystyle M}$ and ${\displaystyle N}$."

If ${\displaystyle M'}$ and ${\displaystyle N'}$ are two other aggregates without common elements, and if ${\displaystyle M\sim M'}$ and ${\displaystyle N\sim N'}$, we saw that we have

Hence the cardinal number of ${\displaystyle (M,N)}$ only depends upon the cardinal numbers ${\displaystyle {\overline {\overline {M}}}={\mathfrak {a}}}$ and ${\displaystyle {\overline {\overline {N}}}={\mathfrak {b}}}$.

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

(1)

${\displaystyle {\mathfrak {a}}+{\mathfrak {b}}=({\overline {\overline {M,N}}})}$.