---
title: Minimality exercise from Halmos
tags: math, halmos, set-theory
status: notes
---
The task given in Halmos's *Naive Set Theory* is to show that given some
subset of a natural number $E$, there is some $k\in E$ such that
for any other $m\in E$, we have $k\in m$. Intuitively, we know
that “$k\in m$” means “$k < m$”, but since we haven't defined
order yet in the book, we have to deal just with the set theoretic
properties of the naturalvnumbers. Now how should we go about finding
this $k$? One way is to look at an example to see what happens.
Suppose we take $E = \{1,3,7\}$. Then the “$k$” that we want
is $1$, since $1\in 3, 1\in 7$.