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Definition

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

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

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

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

Proper subset

A subset is called proper if $A\subseteq B$ and $A\ne B$.

Immediate claims

1. Note that $\emptyset$$\subseteq A$ for all sets $A$.
2. Note that $\emptyset\subset A$ for all non-empty sets $A$.

See also

• Partial order
• Subset relation
• Lattice Theory (subject)

References

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