Number of Elements in Partition

Theorem

Let $S$ be a set.

Let there be a partition on $S$ of $n$ subsets, each of which has $m$ elements.

Then:

$\left|{S}\right| = n m$

Proof

Let the partition of $S$ be $S_1, S_2, \ldots, S_n$.

Then:

$\forall k \in \left[{1 \,.\,.\, n}\right]: \left|{S_k}\right| = m$

By Power of an Element:

$\displaystyle \sum_{k=1}^n \left|{S_k}\right| = n m$

and the result follows from the Fundamental Principle of Counting.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 19$: Theorem $19.1$