Textbook
The universal set is $U$, with a subset $A\subseteq U$. The complement of $A$ relative to $U$ is denoted $A^c$ and defined by:
$$A^c:=U\setminus A=\{x\in U\mid x\notin A\}.$$
The universe is part of the definition. Changing $U$ can change the complement while $A$ stays the same. The notation $A^c$ applies only where $U$ has already been specified.
Within this fixed $U$, the identities hold: $A\cap A^c=\varnothing$ and $A\cup A^c=U$: each element of the universe either belongs to $A$ or does not. Taking the complement twice returns the original set: $(A^c)^c=A$, because for $x\in U$, denying non-membership is equivalent to membership.
The complement of the empty set is $U$, and the complement of $U$ is empty. These statements concern the chosen universe, not an unrestricted “everything in existence.”
Membership in a complement has two requirements: the object is in the universe and absent from $A$. A negative answer about $A$ alone is insufficient. Within a fixed universe, complementation reverses the membership answer for every element. Reversing those answers twice restores them. If the universe changes between the two steps, the double-complement argument no longer concerns the same operation: the collection of objects whose answers are compared has changed.
Related concepts: Set difference.
In plain language
Complement resembles the list of absent people among those invited to a meeting. The invitation list fixes the scope, and attendees are removed from it. Without that scope, almost the entire planet enters the minutes and the meeting drowns in its own paperwork. An uninvited person does not count as an absent invitee; changing the initial list changes the answer.
Example
In this illustrative model, $U=\{1,2,3,4,5\}$ and $A=\{1,2,3\}$. The final step shows the effect of a different universe.
Set complement
Fixed universe
Only these five listed elements are under consideration.
Step 1 of 3: Fixed universe
Read all steps
Fixed universe
Only these five listed elements are under consideration.
- U
- 1current2current3current4current5current
Exclusion of A
The elements 1, 2, and 3 belong to A. The complement retains 4 and 5.
- U
- 1excluded2excluded3excluded4included5included
- Complement
- 4included5included
Changed universe
With U = {1,2,3,4}, the number 5 is no longer considered. The same A now has a different complement.
- New U
- 1excluded2excluded3excluded4included
- New complement
- 4included
In the first universe, four and five pass both checks: they are listed in $U$ and absent from $A$. Six is also absent from $A$, but cannot be added because it lies outside $U$. In the last step, five disappears from the domain of consideration. It has not entered $A$ or changed its properties. A common mistake is to carry an old complement into a new problem without checking the new universe.