Chứng minh:

a) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C);

b) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

Lời giải

a) Chiều thuận:

Xét x ∈ A ∩ (B ∪ C).

⇒ x ∈ A và x ∈ (B ∪ C).

\[ \Rightarrow \left\{ \begin{array}{l}x \in A\\\left[ \begin{array}{l}x \in B\\x \in C\end{array} \right.\end{array} \right. \Rightarrow \left[ \begin{array}{l}\left\{ \begin{array}{l}x \in A\\x \in B\end{array} \right.\\\left\{ \begin{array}{l}x \in A\\x \in C\end{array} \right.\end{array} \right. \Rightarrow x \in \left( {A \cap B} \right) \cup \left( {A \cap C} \right)\]   (1)

Chiều đảo:

Xét x ∈ (A ∩ B) ∪ (A ∩ C).

⇒ x ∈ (A ∩ B) hoặc x ∈ (A ∩ C).

\( \Rightarrow \left[ \begin{array}{l}\left\{ \begin{array}{l}x \in A\\x \in B\end{array} \right.\\\left\{ \begin{array}{l}x \in A\\x \in C\end{array} \right.\end{array} \right. \Rightarrow \left\{ \begin{array}{l}x \in A\\\left[ \begin{array}{l}x \in B\\x \in C\end{array} \right.\end{array} \right. \Rightarrow x \in A \cap \left( {B \cup C} \right)\)   (2)

Từ (1), (2), suy ra điều phải chứng minh.

b) Chiều thuận:

Xét x ∈ A ∪ (B ∩ C).

⇒ x ∈ A hoặc x ∈ (B ∩ C).

\( \Rightarrow \left[ \begin{array}{l}x \in A\\\left\{ \begin{array}{l}x \in B\\x \in C\end{array} \right.\end{array} \right. \Rightarrow \left\{ \begin{array}{l}\left[ \begin{array}{l}x \in A\\x \in B\end{array} \right.\\\left[ \begin{array}{l}x \in A\\x \in C\end{array} \right.\end{array} \right. \Rightarrow x \in \left( {A \cup B} \right) \cap \left( {A \cup C} \right)\)    (3)

Chiều đảo:

Xét x ∈ (A ∪ B) ∩ (A ∪ C).

⇒ x ∈ (A ∪ B) và x ∈ (A ∪ C).

\( \Rightarrow \left\{ \begin{array}{l}\left[ \begin{array}{l}x \in A\\x \in B\end{array} \right.\\\left[ \begin{array}{l}x \in A\\x \in C\end{array} \right.\end{array} \right. \Rightarrow \left[ \begin{array}{l}x \in A\\\left\{ \begin{array}{l}x \in B\\x \in C\end{array} \right.\end{array} \right. \Rightarrow x \in A \cup \left( {B \cap C} \right)\)    (4)