Question

hello ...iam looking for proof of "demorgen's theroems" in digital ,,,,can you help me ?!

Answer 1

Proposition: (A∩B) ′ ⊆(A ′ ∪B ′ )

Proof: (by contradiction)
Assume: (A∩B) ′ ⊈(A ′ ∪B ′ )
∴∃x such that x∈(A∩B) ′ , x∉(A ′ ∪B ′ )
[x∉(A ′ ∪B ′ )]⟹(x∉A ′ ∧x∉B ′ )
⟹(x∈A∧x∈B)
⟹[x∈(A∩B)]
⟹[x∉(A∩B) ′ ]
⊕ (contradiction)
∴(A∩B) ′ ⊆(A ′ ∪B ′ ) … (1) Proposition: (A ′ ∪B ′ )⊆(A∩B) ′

Proof: (by contradiction)
Assume: (A ′ ∪B ′ )⊈(A∩B) ′
∴∃x such that x∉(A∩B) ′ , x∈(A ′ ∪B ′ )
[x∉(A∩B) ′ ]⟹[x∈(A∩B)]
⟹(x∈A∧x∈B)
⟹[x∉A ′ ∧x∉B ′ )]
⟹[x∉(A ′ ∪B ′ )]
⊕ (contradiction)
∴(A ′ ∪B ′ )⊆(A∩B) ′ … (2) By (1) and (2), (A ′ ∪B ′ )=(A∩B) ′