De Morgans law
De Morgans law : The complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements.These are called De Morgan’s laws.These are named after the mathematician De Morgan.
The laws are as follows :
( A ∪ B)^{’} = A^{’} ∩ B^{’} ( A∩B)^{’} = A^{’} ∪ B^{’} |
Examples on De Morgans law :
1) Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}.
Show that (A ∪B) ^{ '} = A ^{ '} ∩ B ^{'} .
Solution :
U = {1, 2, 3, 4, 5, 6}
A = {2, 3}
B = {3, 4, 5}
A ∪ B = {2, 3} ∪ {3, 4, 5}
= {2, 3, 4, 5}
∴ (A ∪ B) ^{ '} = {1, 6}
Also A ^{ '} = {1, 4, 5, 6}
B ^{ '} = {1, 2, 6}
∴ A ^{'} ∩ B ^{'} = {1, 4, 5, 6} ∩ {1, 2, 6}
= {1, 6}
Hence (A ∪ B) ^{'} = A ^{ '} ∩ B ^{'}
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2) If ξ = {a,b,c,d,e}, A = { a,b,d} and B = {b,d,e}. Prove De Morgan's law of intersection.
Solution :
ξ = {a,b,c,d,e}
A = { a,b,d}
B = {b,d,e}
(A ∩ B) = { a,b,d} ∩ {b,d,e}
(A ∩ B) = {b,d}
∴ (A ∩ B)' = {a, c,e} ----->(1)
A' = {c,e} and B' = {a,c}
∴ A' ∪ B' = {c,e} ∪ {a,c}
A' ∪ B'= { a, c,e} ----->(2)
From (1) and (2)
(A ∩ B)' = A' ∪ B' (which is a De Morgan's law of intersection).
Set Theory
• Sets
• Representation of Set
• Cardinal Number
• Types of Sets
• Pairs of Sets
• Subset
• Complement of Set
• Union of the Sets
• Intersection of Sets
• Operations on Sets
• De Morgans Law
• Venn Diagrams
• Venn-diagrams for sets
• Venn-diagrams for different situations
• Problems on Intersection of Two Sets
• Problems on Intersection of Three Sets