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The elements of this aggregate are, therefore, the elements of $M$, of $N$, of $P$, $...$, taken together.

We will call by the name "part" or "partial aggregate" of an aggregate $M$ any other aggregate $M_{1}$ whose elements are also elements of $M$.

If $M_{2}$ is a part of $M_{1}$ and $M_{1}$ is a part of $M$, then $M_{2}$ is a part of $M$.

Every aggregate $M$ has a definite "power," which we will also call its "cardinal number."

We will call by the name "power" or "cardinal number" of $M$ the general concept which, by means of our active faculty of thought, arises from the aggregate $M$ when we make abstraction of the nature of its various elements $m$ and of the order in which they are given.

[482] We denote the result of this double act of abstraction, the cardinal number or power of $M$, by

Since every single element $m$, if we abstract from its nature, becomes a "unit," the cardinal number ${\overline {\overline {M}}}$ is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate $M$.

We say that two aggregates $M$ and $N$ are "equivalent," in signs

if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element