Prove A = (A\B) ∪ (A ∩ B) I have to demonstrate this formulae:
Prove $A = (A\setminus B) ∪ (A ∩ B)$
But it seems to me that it is false.

$(A\setminus B) ∪ (A ∩ B)$ 
  
  
*
  
*$X \in A\setminus B \implies  { x ∈ A  \text{ and } x ∉ B }$
  
  
  or                                    
  
  
*
  
*$X ∈ A ∩ B \implies { x ∈ A \text{ and } x ∈ B }$
  



so: 
$x ∈ A ∩ B$
so:
  $A ≠ (A\setminus B) ∪ (A ∩ B)$

Did I solve the problem or I am just blind?

$\rm\ A\backslash B\  =\ A\cap\overline B\ \ \:$ so $\rm\ \: (A\backslash B)\cup (A\cap B)\ =\ (A\cap\overline B)\cup(A\cap B)\ =\ A\cap(\overline B\cup B)\ =\ A$
