Definition. Let $A$ be a non-empty set. A collection $P$ of subsets $A_1,A_2,A_3,\ldots$ of $A$ is called a partition of $A$ if the following three conditions hold:
- $\emptyset \notin P$,
- $A_1 \cup A_2 \cup A_3 \cup \cdots = A$, and
- For all $i$ and $j$, if $i \ne j$, then $A_i \cap A_j = \emptyset$.
Definition. Let $n$ be a positive integer. A partition of $n$ is a finite sequence $\lambda_1,\lambda_2,\lambda_3,\ldots\lambda_k$ of positive integers such that $\lambda_1 \ge \lambda_2 \ge \lambda_3 \ge \cdots \ge \lambda_k$ and $$\lambda_1+\lambda_2+\lambda_3+\cdots+\lambda_k = n.$$
I want to know a derivation of the word "partition" in the integer from the partition of a set. Here's my attempt.
First, define a relation $R$ on the set $\Bbb N$ as follows: For all $a,b \in \Bbb N,$ $$aRb \iff (\exists I \subseteq \Bbb N) (a,b) \in I.$$
Let $n \in \Bbb N$. I have already shown that $R$ is an equivalence relation on $\Bbb N$. Hence, $\Bbb N$ could be partitioned into equivalence classes. Call them $$[\lambda_k, \lambda_{k-1}), \ldots, [\lambda_2, \lambda_1), [\lambda_1, +\infty),$$ where $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_k$ and $\lambda_1+\lambda_2+\cdots+\lambda_k=n$. Let $$P=\{[\lambda_k, \lambda_{k-1}), \ldots, [\lambda_2, \lambda_1), [\lambda_1, +\infty)\}.$$
But, I got stucked here. Any ideas? Thanks in advanced.
I can only see the following vague link: any partition of a positive integer $n$ may be obtained by partitioning a set $E$ of $n$ elements, and taking the number of elements of each subset of $E$ belonging to the partition.