Subset
Calculus & Advanced MathA 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. Think of A as fitting inside B.
Example
$A = \{2, 4\}$ and $B = \{1, 2, 3, 4, 5\}$. Since $2 \in B$ and $4 \in B$, we have $A \subseteq B$. Also, every set is a subset of itself: $B \subseteq B$.
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 $B$.
Definition
$A \subseteq B$ means: for all $x$, if $x \in A$ then $x \in B$. $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$. This gives the standard proof technique for set equality: prove mutual containment.
Example
Prove $A \subseteq A \cup B$: take any $x \in A$. By definition of union, $x \in A \cup B$. Done. Proving $A = B$ always requires showing both $A \subseteq B$ and $B \subseteq A$.
Key Insight
The power set $P(A)$ is the set of all subsets of $A$. If $|A| = n$, then $|P(A)| = 2^n$. This exponential growth explains why even small sets have many subsets.
Definition
The subset relation $\subseteq$ defines a partial order on any collection of sets: it is reflexive ($A \subseteq A$), antisymmetric ($A \subseteq B$ and $B \subseteq A$ implies $A = B$), and transitive ($A \subseteq B$ and $B \subseteq C$ implies $A \subseteq C$). The power set $P(X)$ under $\subseteq$ forms a Boolean lattice.
Example
In topology, a topology on X is a collection of subsets (open sets) satisfying closure under arbitrary union and finite intersection. Every topological concept builds on the subset relation.
Key Insight
The lattice of subsets under $\subseteq$ is the prototypical Boolean algebra, connecting set theory to logic (where $\subseteq$ corresponds to implication) and circuit design.