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 ∪ 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)'

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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