Answer to Question #169987 in Discrete Mathematics for EUGINE HAWEZA

Question

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

Answer 1

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 $A \cap \left( {B \cup C} \right) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right)$ .

The statement is proven