ÁLGEBRA DE CONJUNTOS (MATEMÁTICA ELEMENTAR)
Autor: Aécio | Publicado: 2015-08-07T04:25:00-07:00

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Sejam A, B e C conjuntos. Demonstre as afirmações verdadeiras e dê contraexemplo para as falsas.

(A U B) - C = (A - C) U (B - C)  

Pela hipótese, (A U B) - C ⊂ (A - C) U (B - C) ∧ (A U B) - C&nbsp;⊃ (A - C) U (B - C).

* Provando a inclusão [⊂]:
Seja x ∈ (A U B) – C um número arbitrário, pelas definições temos que&nbsp; x ∈ A ∨&nbsp; x ∈ B ∧&nbsp; x ∉ C →  (x ∈ A ∧&nbsp; x ∉ C) ∨ (x ∈ B ∧ x ∉ C) → x ∈ (A – C ) ∨&nbsp; x ∈ (B – C ) ↔ x ∈ (A - C) U (B - C). Logo,&nbsp; (A U B) - C ⊂ (A - C) U (B - C).

** [⊃]:
Seja, ainda,&nbsp;  x ∈ (A - C) U (B - C) → (x ∈ A ∧&nbsp; x ∉ C) ∨ (x ∈ B ∧ x ∉ C) →  x ∈ A ∨&nbsp; x ∈ B ∧&nbsp; x ∉ C ↔ x ∈ (A U B) – C. 

(Verdadeira)


(A - B) ∩  C = (A ∩  C) -&nbsp; (B ∩  C)&nbsp;  

(A - B) ∩ ⊂ (A ∩  C) -&nbsp; (B ∩  C)&nbsp; ∧&nbsp; (A - B) ∩  C ⊃ (A ∩  C) -&nbsp; (B ∩  C)&nbsp;&nbsp;

* Inclusão [⊂]: 
Sendo x ∈ (A - B) ∩  C →&nbsp; (x ∈ A ∧&nbsp; x ∉ B)&nbsp; ∧ x ∈ C → (x ∈ A ∧&nbsp; x ∈ C) ∧ (x ∉ B ∧ x ∈ C) → x ∈ (A ∩  C) ∧ x ∉ (B ∩  C)&nbsp;↔ x ∈ (A ∩  C) - (B ∩  C).

** [⊃]:
A recíproca é logicamente óbvia.

∴ (A - B) ∩  C = (A ∩  C) -&nbsp; (B ∩  C)&nbsp;