Demorgan's Law of Set Theory Proof

De Morgan's laws are a pair of transformation rules relating the set operators "union" and "intersection" in terms of each other by means of negation.

De Morgans Law of Set Theory Proof - Math Theorems

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 ∪ BDefinition 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 ∩ BDefinition 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)'

De Morgan's laws are a pair of transformation rules relating the set operators "union" and "intersection" in terms of each other by means of negation.

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