Proper Subset
A proper subset is a subset that is strictly smaller than the original set, meaning it is missing at least one element.
Formula
A \subset B: A \subseteq B \text{ and } A \neq B
Definition
A proper subset is a subset that does not use all the elements of the original set; it is truly "inside" the other set with at least one element left out. Formally, $A$ is a proper subset of $B$ (written $A \subset B$ or $A \subsetneq B$) if $A \subseteq B$ and $A \neq B$, equivalently every element of $A$ is in $B$ and there exists at least one element in $B$ not in $A$. In the context of infinite sets, a set can be in bijection with a proper subset of itself (Dedekind-infinite sets): $\mathbb{N} \cong \{\text{even numbers}\}$ via $n \mapsto 2n$, often used as the very definition of an infinite set.
Example
$\{1, 2\} \subset \{1, 2, 3\}$ because $\{1, 2\}$ is a subset and there is an extra element ($3$) in the larger set, but $\{1, 2, 3\}$ is not a proper subset of itself. The subsets of $\{a, b\}$ are $\emptyset$, $\{a\}$, $\{b\}$, $\{a,b\}$; the proper subsets are $\emptyset$, $\{a\}$, $\{b\}$. Hilbert's Hotel illustrates the infinite case: a countably infinite hotel can accommodate new guests even when "full" by shifting all current guests, showing the hotel is Dedekind-infinite.
Key Insight
Every proper subset is a subset, but not every subset is proper; a set is a subset of itself but never a proper subset of itself. Cantor's theorem states the power set $P(A)$ is always a proper superset of $A$ in cardinality, $|P(A)| > |A|$, showing there is no largest infinity, and the existence of a bijection between a set and a proper subset characterizes infinite sets in Dedekind's formulation, an alternative to cardinality-based definitions of infinity.