Proper Subset¶

Definition¶

Let $A$ and $B$ be sets. We say that $A$ is a proper subset of $B\text{,}$ denoted unambiguously by $A\subsetneq B\text{,}$ if every element of $A$ is also an element of $B\text{,}$ but some element of $B$ is not in $A\text{.}$ Equivalently,

Interpretation¶

A proper subset captures strict inclusion between sets: $A\subsetneq B$ means that $A$ lies inside $B$ and $B$ has at least one element left over. It distinguishes containment from equality.

In the left panel the disc $A$ lies inside $B$, and the marked point $y$ belongs to $B$ but not to $A$. This one point, $y\in B\setminus A$$A$, is what makes the inclusion strict. In the right panel the two discs coincide: $A$ is still a subset of $B$, but not a proper one.

The definition has two parts, and so does a proof: show $A\subseteq B$$B$, then name one witness in $B\setminus A$$A$. For example, $\mathbb{N}^+\subsetneq\mathbb{N}$ with witness $0$, and the empty set is a proper subset of every nonempty set. For finite sets, a proper subset has fewer elements: $|A|<|B|$$|B|$.

Notation is a common source of confusion. Some texts write $A\subset B$ for $A\subseteq B$$B$, others for $A\subsetneq B$$B$. The Definition therefore uses $\subsetneq$, which has only one reading; when a source writes $\subset$, check which meaning it intends.

Key Results¶

• Every proper subset is a subset, but not conversely.
• Proper inclusion is transitive, i.e., $A\subsetneq B$ and $B\subsetneq C$ implies $A\subsetneq C\text{.}$

Examples¶

• $\{1,\,2\}\subsetneq\{1,\,2,\,3\}\text{.}$
• A set is never a proper subset of itself.

Prerequisites¶

• Subset

Related Concepts¶

• Empty Set
• Difference Set
• Singleton Set

Sources¶

• Math I: Calculus · Lecture 1

Aliases: strict subset Symbols: element of, natural numbers, subset, there exists

Last updated: 22 October 2026 19:04