# How to prove this? 6) For sets A,B,C prove A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) by showing Left side ⊆ Right side and Right side ⊆ Left side.

Feb 24, 2018

Proof:-$\text{ } A \cup \left(B \cap C\right) = \left(A \cup B\right) \cap \left(A \cup C\right)$

Let,
$\text{ } x \in A \cup \left(B \cap C\right)$

$\implies x \in A \vee x \in \left(B \cap C\right)$

$\implies x \in A \vee \left(x \in B \wedge x \in C\right)$

$\implies \left(x \in A \vee x \in B\right) \wedge \left(x \in A \vee x \in C\right)$

$\implies x \in \left(A \cup B\right) \wedge x \in \left(A \cup C\right)$

$\implies x \in \left(A \cup B\right) \cap \left(A \cup C\right)$

• $x \in A \cup \left(B \cap C\right) \implies x \in \left(A \cup B\right) \cap \left(A \cup C\right)$

=>color(red)(Auu(BnnC)sube(AuuB)nn(AuuC)

Let,
$\text{ } y \in \left(A \cup B\right) \cap \left(A \cup C\right)$

$\implies y \in \left(A \cup B\right) \wedge y \in \left(A \cup C\right)$

$\implies \left(y \in A \vee y \in B\right) \wedge \left(y \in A \vee y \in C\right)$

$\implies y \in A \vee \left(y \in B \wedge y \in C\right)$

$\implies y \in A \vee y \in \left(B \cap C\right)$

$\implies y \in A \cup \left(B \cap C\right)$

• $x \in \left(A \cup B\right) \cap \left(A \cup C\right) \implies x \in A \cup \left(B \cap C\right)$

=>color(red)((AuuB)nn(AuuC)subeAuu(BnnC)

From the both red part , we get by using the rule of equal set,

color(red)(ul(bar(|color(green)(Auu(BnnC)=(AuuB)nn(AuuC)))|

Hope it helps...
Thank you...