Answer to Question #157250 in Real Analysis for David

Question

Prove A∩(B∩C)=(A∩B)∩C

Answer 1

Let$x \in A \cap (B \cap C)$

$\implies x \in A$ and $x \in B \cap C$

$\implies x \in A$ and $x \in B$ and $x \in C$

$\implies x\in A \cap B$ and $x \in C$

$\implies x \in (A\cap B) \cap C$

$\implies A\cap (B \cap C) \sub (A\cap B) \cap C$

Conversely, let $y \in (A\cap B) \cap C$

$\implies y \in A \cap B$ and $y \in C$

$\implies y \in A$ and $y \in B$ and $y \in C$

$\implies y\in A$ and $y\in B\cap C$

$\implies y\in A \cap (B\cap C)$

$\implies (A\cap B) \cap C \sub A \cap (B \cap C)$

Hence, $A \cap (B \cap C) = (A\cap B) \cap C$