Proper subset

A subset that is strictly smaller than the containing set.

Proper subset

A proper subset of a set $B$ is a subset $A$ of $B$ that is not equal to $B$ . Formally,

A \subsetneq B \quad\text{means}\quad (A\subseteq B)\ \land\ (A\neq B).

Proper inclusion $\subsetneq$ indicates strict containment. In analysis it is used, for example, to describe dense proper subsets (e.g., $\mathbb{Q}\subsetneq\mathbb{R}$ ) and strict refinements of covers or partitions.

Examples:

$\mathbb{Q}\subsetneq \mathbb{R}$ .
$\varnothing \subsetneq \{0\}$ .
$\{0\}\not\subsetneq \{0\}$ (it is a subset but not a proper subset).