Complement of a Set
The complement of a set $A$ contains all elements in the universal set that are NOT in $A$, written $A'$ or $A^c$.
Formula
A' = U \setminus A = \{x \in U : x \notin A\}
Definition
The complement of a set $A$ is everything in the universe that is not in $A$; if you know what is included, the complement is everything left out. Formally, $A'$ (or $A^c$) $= \{x \in U : x \notin A\}$, with key properties $(A')' = A$, $A \cup A' = U$, $A \cap A' = \emptyset$, and De Morgan's laws $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$. In measure theory, the complement of a measurable set is measurable, and sigma-algebras are closed under complements; in Boolean algebra, complementation is the involution satisfying $a \wedge a' = 0$ and $a \vee a' = 1$, and in topology closed sets are complements of open sets.
Example
If $U = \{1,2,3,4,5\}$ and $A = \{1,3,5\}$, then $A' = \{2,4\}$: what was in $A$ is now out, and what was out is now in. In probability, $P(A') = 1 - P(A)$: if the probability of rain is $0.3$, the probability of no rain is $0.7$. The set of irrational numbers is the complement of $\mathbb{Q}$ in $\mathbb{R}$; it is measurable in the Lebesgue sigma-algebra, with measure $1$ on $[0,1]$.
Key Insight
Think of a complement as the "everything else" set; in a Venn diagram it is the region outside the circle but inside the rectangle (universe). De Morgan's laws are essential for logic circuits and database queries: negating a union becomes an intersection of negations. Closure under complementation distinguishes sigma-algebras from simpler set systems, precisely the property needed to define probability for all "observable" events.