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:

$\card S = n m$

Proof

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

Then:

$\forall k \in \set {1, 2, \ldots, n}: \card {S_k} = m$

By definition of multiplication:

$\ds \sum_{k \mathop = 1}^n \card {S_k} = n m$

and the result follows from the Fundamental Principle of Counting.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 19$: Combinatorial Analysis: Theorem $19.1$