Proper subset

A strict inclusion A ⊊ B, meaning A ⊆ B but A ≠ B

Proper subset

Let $A$ and $B$ be sets . We say that $A$ is a proper subset of $B$ , written $A \subsetneq B$ , if

A \subseteq B \quad\text{and}\quad A \neq B,

where $\subseteq$ is the subset relation.

Proper subset expresses strict containment, in contrast to $A \subseteq B$ which allows equality. In particular, the empty set is a proper subset of any nonempty set.

Examples:

$\{1\} \subsetneq \{1,2\}$ .
$\emptyset \subsetneq \{0\}$ .
$\{1,2\} \not\subsetneq \{1,2\}$ (it is not strict).