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Class 11 | CBSE | MATHEMATICS
SETS | COMPLEMENT OF SETS | De-Morgan's Law
De Morgan’s Laws & Complement of Sets
Let A and B be any two subsets of the universal set
U.
Then the following important relations hold between
union,
intersection and
complement.
De Morgan’s Laws
Mathematical Form:
(A ∪ B)′ = A′ ∩ B′
(A ∩ B)′ = A′ ∪ B′
In words:
- The complement of the union of two sets is the intersection of their complements.
- The complement of the intersection of two sets is the union of their complements.
These results are known as De Morgan’s Laws, named after the
mathematician Augustus De Morgan.
The complement A′ of a set A is represented in a Venn diagram
by shading the region of the universal set U that lies
outside A.
The shaded region represents the complement of the set.
📌 Some Properties of Complement Sets
1. Complement Laws
- A ∪ A′ = U
- A ∩ A′ = φ
2. De Morgan’s Laws
- (A ∪ B)′ = A′ ∩ B′
- (A ∩ B)′ = A′ ∪ B′
3. Law of Double Complementation
(A′)′ = A
4. Laws of Empty Set and Universal Set
- φ′ = U
- U′ = φ
🧠 Memory Hook:
De Morgan’s Laws simply say:
“Complement flips AND ↔ OR”
and moves the complement inside the brackets.
HOME BUTTONS
The shaded region represents the complement of the set.
- A ∪ A′ = U
- A ∩ A′ = φ
- (A ∪ B)′ = A′ ∩ B′
- (A ∩ B)′ = A′ ∪ B′
(A′)′ = A
- φ′ = U
- U′ = φ