Complement of a Set

The complement of a set $A$ is the set of all elements in the universal set $U$ that are not in $A$.

• Notation: $A^c$ or sometimes $\bar{A}$
• Definition: $A^c=U\text{A}=\{x\in U\mid x\notin A\}$
• Example:
  • Let $U=\{1,2,3,4,5\}$ and $A=2,4$
  • Then $A^c=\{1,3,5\}$
• Properties:
  • $A\cap A^c=\varnothing$ (no overlap with its complement)
  • $A\cup A^c=U$ (together they cover the universal set)