Subset
A subset is a set whose every element is also contained in another set, written A ⊆ B.
Formula
A \subseteq B: \text{ every element of } A \text{ is in } B
Definition
Set $A$ is a subset of set $B$ if every element in $A$ is also in $B$, so $A$ fits inside $B$. Formally, $A \subseteq B$ means: for all $x$, if $x \in A$ then $x \in B$; $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$, the standard proof technique for set equality. The subset relation $\subseteq$ defines a partial order on any collection of sets: reflexive, antisymmetric, and transitive, and the power set $P(X)$ under $\subseteq$ forms a Boolean lattice.
Example
For $A = \{2, 4\}$ and $B = \{1, 2, 3, 4, 5\}$: since $2 \in B$ and $4 \in B$, $A \subseteq B$; every set is also a subset of itself. Proving $A \subseteq A \cup B$: take any $x \in A$, then by definition of union, $x \in A \cup B$. In topology, a topology on $X$ is a collection of subsets (open sets) satisfying closure under arbitrary union and finite intersection, so every topological concept builds on the subset relation.
Key Insight
The empty set $\emptyset$ is a subset of every set, because there are no elements in $\emptyset$ that could possibly fail to be in $B$. The power set $P(A)$, the set of all subsets of $A$, has $|P(A)| = 2^n$ when $|A| = n$, explaining why even small sets have many subsets. The lattice of subsets under $\subseteq$ is the prototypical Boolean algebra, connecting set theory to logic (where $\subseteq$ corresponds to implication) and circuit design.