# Subset of

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## Definition

A set [ilmath]A[/ilmath] is a subset of a set [ilmath]B[/ilmath] if [ilmath]A\subseteq B[/ilmath], that is (by the implies-subset relation):

• [ilmath]\forall a\in A[a\in B][/ilmath] which comes from [ilmath]\forall x[x\in A\implies x\in B][/ilmath].

This may be written [ilmath]A\in\mathcal{P}(B)[/ilmath] (where [ilmath]\mathcal{P}(B)[/ilmath] denotes the power-set of [ilmath]B[/ilmath], by definition, all subsets of [ilmath]B[/ilmath]!), or [ilmath]A\subseteq B[/ilmath].

Some authors use [ilmath]A\subset B[/ilmath], however we reserve this for proper subset - see below. Some authors who use this for what we use [ilmath]\subseteq[/ilmath] use [ilmath]\subsetneq[/ilmath] for the proper case.

### Proper subset

A subset is called proper if [ilmath]A\subseteq B[/ilmath] and [ilmath]A\ne B[/ilmath].

## Immediate claims

1. Note that [ilmath]\emptyset[/ilmath][ilmath]\subseteq A[/ilmath] for all sets [ilmath]A[/ilmath].
2. Note that [ilmath]\emptyset\subset A[/ilmath] for all non-empty sets [ilmath]A[/ilmath].