Sets | Class 11 Mathematics Notes

1.10 Complement of a Set

Given a universal set U and a subset A of U, the complement of A, denoted A', is the set of all elements in U that are not in A. Formally, A' = {x : x ∈ U and x ∉ A}. The complement can also be viewed as the difference U - A. For example, if U is the set of prime numbers and A is the set of prime numbers not dividing 42, then A' is the set of prime divisors of 42. Complementation follows important laws: A ∪ A' = U, A ∩ A' = ∅, and double complement (A')' = A. De Morgan's laws relate complements with unions and intersections: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'. Venn diagrams visually represent complements as the shaded region outside the set A within U. Understanding complements is essential for solving complex set problems and proving set identities.

📊 Diagram: Fig 1.10

🧪 Activity: Exercise 1.5 involves finding complements and verifying De Morgan's laws.

🔗 Connection: This section concludes the main operations on sets and leads to summary and miscellaneous examples.