De Morgans law

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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
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