# Subset Relation is Ordering

## Contents

## Theorem

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$ be any subset of $\mathcal P \left({S}\right)$, that is, an arbitrary set of subsets of $S$.

Then $\subseteq$ is an ordering on $\mathbb S$.

In other words, let $\left({\mathbb S, \subseteq}\right)$ be the relational structure defined on $\mathbb S$ by the relation $\subseteq$.

Then $\left({\mathbb S, \subseteq}\right)$ is an ordered set.

### General Result

Let $\mathbb S$ be a set of sets or class.

Then $\subseteq$ is an ordering on $\mathbb S$.

In other words, let $\left({\mathbb S, \subseteq}\right)$ be the relational structure defined on $\mathbb S$ by the relation $\subseteq$.

Then $\left({\mathbb S, \subseteq}\right)$ is an ordered set.

## Proof

To establish that $\subseteq$ is an ordering, we need to show that it is reflexive, antisymmetric and transitive.

So, checking in turn each of the criteria for an ordering:

### Reflexivity

\(\displaystyle \forall T \in \mathbb S: \ \ \) | \(\displaystyle T\) | \(\subseteq\) | \(\displaystyle T\) | $\quad$ Set is Subset of Itself | $\quad$ |

So $\subseteq$ is reflexive.

$\Box$

### Antisymmetry

\(\displaystyle \forall S_1, S_2 \in \mathbb S: \ \ \) | \(\displaystyle S_1 \subseteq S_2 \land S_2 \subseteq S_1\) | \(\iff\) | \(\displaystyle S_1 = S_2\) | $\quad$ Definition of Set Equality | $\quad$ |

So $\subseteq$ is antisymmetric.

$\Box$

### Transitivity

\(\displaystyle \forall S_1, S_2, S_3 \in \mathbb S: \ \ \) | \(\displaystyle S_1 \subseteq S_2 \land S_2 \subseteq S_3\) | \(\implies\) | \(\displaystyle S_1 \subseteq S_3\) | $\quad$ Subset Relation is Transitive | $\quad$ |

That is, $\subseteq$ is transitive.

$\Box$

So we have shown that $\subseteq$ is an ordering on $\mathbb S$.

$\blacksquare$

## Sources

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.5$: Relations

- 1967: Garrett Birkhoff:
*Lattice Theory*(3rd ed.) ... (next): $\S \text I.1$