Page:Cantortransfinite.djvu/114

for all ${\displaystyle n}$'s which are different from ${\displaystyle n_{0}}$.

The totality of different coverings of N with M forms a definite aggregate with the elements ${\displaystyle f(N)}$; we call it the "covering-aggregate (Belegungsmenge) of ${\displaystyle N}$ with ${\displaystyle M}$" and denote it by ${\displaystyle (N|M)}$. Thus:

(2)

${\displaystyle (N|M)={f(N)}}$.

If ${\displaystyle M\sim M'}$ and ${\displaystyle N\sim N'}$, we easily find that

(3)

${\displaystyle (N|M)\sim (N'|M')}$.

Thus the cardinal number of ${\displaystyle (N|M)}$ depends only on the cardinal numbers ${\displaystyle {\overline {\overline {M}}}={\mathfrak {a}}}$ and ${\displaystyle {\overline {\overline {N}}}={\mathfrak {b}}}$; it serves us for the definition of ${\displaystyle {\mathfrak {a}}^{\mathfrak {b}}}$ :

(4)

${\displaystyle {\mathfrak {a}}^{\mathfrak {b}}=({\overline {\overline {N|M}}})}$.

For any three aggregates, ${\displaystyle M,N,P}$, we easily prove the theorems:

(5)

${\displaystyle ((N|M).(P|M))\sim ((N.P)|M)}$,

(6)

${\displaystyle ((P|M).(P|N))\sim (P|(M.N))}$,

(7)

${\displaystyle (P|(N|M))\sim ((P.N)|M)}$,

from which, if we put ${\displaystyle {\overline {\overline {P}}}={\mathfrak {c}}}$, we have, by (4) and by paying attention to § 3, the theorems for any three cardinal numbers, ${\displaystyle {\mathfrak {a}}}$, ${\displaystyle {\mathfrak {b}}}$, and ${\displaystyle {\mathfrak {c}}}$:

(8)

${\displaystyle {\mathfrak {a}}^{\mathfrak {b}}.{\mathfrak {a}}^{\mathfrak {c}}={\mathfrak {a}}^{\mathfrak {b+c}}}$,

(9)

${\displaystyle {\mathfrak {a}}^{\mathfrak {c}}.{\mathfrak {b}}^{\mathfrak {c}}=({\mathfrak {a}}.{\mathfrak {b}})^{\mathfrak {c}}}$,

(10)

${\displaystyle ({\mathfrak {a}}^{\mathfrak {b}})^{\mathfrak {c}}={\mathfrak {a}}^{\mathfrak {b.c}}}$.