Subset¶

Definition¶

Let $A\text{,}$ $B$ be sets. We say that $A$ is a subset of $B\text{,}$ denoted $A\subseteq B\text{,}$ if every element of $A$ is also an element of $B\text{,}$ i.e.,

Interpretation¶

A subset describes containment between collections of objects: $A\subseteq B$ means that whatever belongs to $A$ also belongs to $B$. The definition allows $A=B$$B$.

In the left panel the disc $A$ lies entirely inside $B$, so every $x\in A$ is also in $B$. In the right panel $A$ sticks out of $B$, and the single marked point $x\in A$ with $x\notin B$ is enough to show that $A\not\subseteq B$$B$. To prove $A\subseteq B$$B$, take an arbitrary $x\in A$ and show $x\in B$$B$; to disprove it, one point of $A$ outside $B$ suffices.

Since the definition is an implication, there is nothing to check when $A=\varnothing$$\varnothing$, which is why the empty set is a subset of every set. Inclusion in both directions gives equality: $A\subseteq B$ and $B\subseteq A$ together mean $A=B$$B$. A proper subset adds the requirement $A\neq B$$B$.

A common slip is to expect any two sets to be comparable, as any two numbers are under $\le$: neither $\{1\}\subseteq\{2\}$ nor $\{2\}\subseteq\{1\}$ holds.

Key Results¶

• Every set is a subset of itself.
• Subset relations are transitive, i.e., $A\subseteq B$ and $B\subseteq C$ implies $A\subseteq C\text{.}$
• Proper subsets exclude equality.

Examples¶

• $\{1,\,2\}\subseteq\{1,\,2,\,3\}\text{.}$
• $\varnothing$ is a subset of every set.

Prerequisites¶

• Set

Related Concepts¶

• Empty Set
• Union of Sets

Sources¶

• Math I: Calculus · Lecture 1

Symbols: element of, subset

Last updated: 20 October 2026 03:02