06-02-2016, 03:02 PM
Statement:
Demorgan's First Law:
(A ∪ B)' = (A)' ∩ (B)'
The first law states that the complement of the union of two sets is the intersection of the complements.
Proof :
(A ∪ B)' = (A)' ∩ (B)'
Consider x ∈ (A ∪ B)'
If x ∈ (A ∪ B)' then x ∉ A ∪ B Definition of compliment
(x ∈ A ∪ B)' Definition ∉
(x ∈ A ∪ x ∈ B)' Definition of ∪
(x ∈ A)' ∩ (x ∈ B)'
(x ∉ A) ∩ (x ∉ B) Definition of ∉
(x ∈ A') ∩ (x ∈ B') Definition of compliment
x ∈ A' ∩ B' Definition of ∩
Therefore,
(A ∪ B)' = (A)' ∩ (B)'
Demorgan's Second Law:
(A ∩ B)' = (A)' ∪ (B)'
The second law states that the complement of the intersection of two sets is the union of the complements.
Proof :
(A ∩ B)' = (A)' ∪ (B)'
Consider x ∈ (A ∩ B)'
If x ∈ (A ∩ B)' then x ∉ A ∩ B Definition of compliment
(x ∈ A ∩ B)' Definition of ∉
(x ∈ A ∩ x ∈ B)' Definition of ∩
(x ∈ A)' ∪ (x ∈ B)'
(x ∉ A) ∪ (x ∉ B) Definition of ∉
(x ∈ A') ∪ (x ∈ B') Definition of compliment
x ∈ A' ∪ B' Definition of ∪
Therefore,
(A ∩ B)' = (A)' ∪ (B)'