did I solves this sets correctly? need to solve this only with what we know about 'algebra of sets', is everythink I did is legal and it's correct?
i. (A1$\bigcup$A2) $\setminus$ (B1$\bigcap$B2) = (A1 $\setminus$ B1)$\bigcup$(A1 $\setminus$ B2)$\bigcup$(A2 $\setminus$ B1)$\bigcup$(A2 $\setminus$ B2)
my answer: 
1.(A1$\bigcup$A2) $\setminus$ (B1$\bigcap$B2) = (A1$\bigcup$A2)$\bigcap$(B1$\bigcap$B2)$\complement$
= (A1$\bigcup$A2)$\bigcap$(B1$\complement$$\bigcup$B2$\complement$)
= (A1$\bigcap$B1$\complement$)$\bigcup$(A1$\bigcap$B2$\complement$)$\bigcup$(A2$\bigcap$B1$\complement$)$\bigcup$(A2$\bigcap$B2$\complement$)
2.(A1 $\setminus$ B1)$\bigcup$(A1 $\setminus$ B2)$\bigcup$(A2 $\setminus$ B1)$\bigcup$(A2 $\setminus$ B2) = (A1$\bigcap$B1$\complement$)$\bigcup$(A1$\bigcap$B2$\complement$)$\bigcup$(A2$\bigcap$B1$\complement$)$\bigcup$(A2$\bigcap$B2$\complement$)
ii. A$\bigoplus$B = A$\complement$$\bigoplus$B$\complement$
1.A$\bigoplus$B = (A $\setminus$ B) $\bigcup$ (B $\setminus$ A) = (A$\bigcap$B$\complement$)$\bigcup$(B$\bigcap$A$\complement$)
2.A$\complement$$\bigoplus$B$\complement$ = (A$\complement$ $\setminus$ B$\complement$)$\bigcup$(B$\complement$ $\setminus$A$\complement$) = (A$\complement$$\bigcap$B)$\bigcup$(B$\complement$$\bigcap$A) = (B$\bigcap$A$\complement$)$\bigcup$(A$\bigcap$B$\complement$) = (A$\bigcap$B$\complement$)$\bigcup$(B$\bigcap$A$\complement$)
 A: Yes, you did just fine. It seems you took each side of the equations to find a common "middle ground." It is clear you have mastered the relevant definitions and properties, such as the definitions of "setminus", "complement", and applications of DeMorgan's and the distributive laws.
FYI: Another way you can prove a set identity: say $A = B$ is to take, e.g., $A$, and show
$$\begin{align} A &= \cdots \\ \\ 
& = \cdots\\ \\ 
& = \cdots \\ \\ 
& = B\end{align}$$
So, in the first identity, for example, you could have argued:
$$\begin{align} \color{blue}{\bf (A_1 \cup A_2)\setminus (B_1 \cap B_2)} & = (A_1\cup A_2) \cap (B_1 \cap B_2)^\complement \\ \\ 
& = (A_1 \cup A_2) \cap (B_1^\complement \cup B_2^\complement)\\ \\
& = (A_1\cap B_1^\complement)\cup (A_1\cap B_2^\complement)\cup (A_2\cap B_1^\complement)\cup (A_2\cap B_2^\complement)\\ \\
&\color{blue}{\bf = (A_1\setminus B_1) \cup (A_1 \setminus B_2) \cup (A_2\setminus B_1) \cup (A_2\setminus B_2)}\end{align}$$
This shows that the left hand side (first line) is equal to the right-hand side (last line).
