a. Let A and B and C be sets, prove that A∩(BUC) = (A∩B)U( A∩C).

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Let $x \in A \cap \left( {B \cup C} \right)$ , then $x \in A$ and $x \in B \cup C$ . Then $x \in A$ and $x \in B$ or $x \in C$ . Then $x \in A$ and $x \in B$ or $x \in A$ and $x \in C$ , but then $x \in \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$ , from where $A \cap \left( {B \cup C} \right) \subset \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$

Let $x \in \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$ . Then $x \in A$ and $x \in B$ or $x \in A$ and $x \in C$ . Then $x \in A$ and $x \in B$ or $x \in C$ , then $x \in A$ and $x \in B \cup C$ , but then $x \in A \cap \left( {B \cup C} \right)$ , from where $A \cap \left( {B \cup C} \right) \supset \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$ .

Since $A \cap \left( {B \cup C} \right) \subset \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$ and $A \cap \left( {B \cup C} \right) \supset \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$ then $A \cap \left( {B \cup C} \right) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$ .

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a. Let A and B and C be sets, prove that A∩(BUC) = (A∩B)U( A∩C).

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Question ID: mtid-5-stid-8-sqid-3324-qpid-2023