Complement of a Set
Calculus & Advanced MathThe 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.
Example
If $U = \{1,2,3,4,5\}$ and $A = \{1,3,5\}$, then $A' = \{2,4\}$. The complement flips membership: what was in $A$ is now out, and what was out is now in.
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).
Definition
$A'$ (or $A^c$) $= \{x \in U : x \notin A\}$. Key properties: $(A')' = A$, $A \cup A' = U$, $A \cap A' = \emptyset$. De Morgan's laws: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.
Example
In probability: $P(A') = 1 - P(A)$. If the probability of rain is $0.3$, the probability of no rain is $0.7$.
Key Insight
De Morgan's laws are essential for logic circuits and database queries: negating a union becomes an intersection of negations.
Definition
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 the complementation laws ($a \wedge a' = 0$ and $a \vee a' = 1$). In topology, closed sets are complements of open sets.
Example
The set of irrational numbers is the complement of $\mathbb{Q}$ in $\mathbb{R}$. It is not measurable in the Borel sigma-algebra from irrationals alone, but measurable in the Lebesgue sigma-algebra with measure $1$ on $[0,1]$.
Key Insight
Closure under complementation distinguishes sigma-algebras from simpler set systems, and is precisely the property needed to define probability for all "observable" events.