How to prove that $B = (B \cap A) \cup (B \cap A^c)$? So, my professor asked us to prove these statements.
$$
B = (B \cap A) \cup (B \cap A^c)\\
A \cup B = A \cup (B \cap A^c)
$$
I understand the meaning behind these, but I'm completely lost as to how to mathematically prove them.
Any help would be appreciated! 
 A: You should prove two containments:


*

*$B \subset (B ∩ A) ∪ (B ∩ A^c)$ Choose any $x \in B$. Now there  are two cases: $x \in A$ or $x \not\in A$. If $x \in A$ then $x \in B \cap A$, so $x \in (B ∩ A) ∪ (B ∩ A^c)$. If  $x \not\in A$, then $x \in B \cap A^c$, so also $x \in (B ∩ A) ∪ (B ∩ A^c)$. Finally $x \in (B ∩ A) ∪ (B ∩ A^c)$.

*$(B ∩ A) ∪ (B ∩ A^c) \subset B$ Choose any $x \in (B ∩ A) ∪ (B ∩ A^c)$. Now you also have two cases: $x \in (B ∩ A)$ or $x \in (B ∩ A^c)$. If $x \in (B ∩ A)$, then $x \in B$, if $x \in (B ∩ A^c)$ then also $x \in B$. Finally $x \in B$.


Can you prove the second this way?
A: $$(B \cap A) \cup (B \cap A^c)=B \cap (A\cup A^c )=B \cap U = B$$  
$$A \cup (B \cap A^c)=  (A \cup B)\cap (A\cup A^c) =(A \cup B)\cap U=(A \cup B) $$
A: Hint: An equality between sets is proved by considering an element in one of the sides and proving (via definitions, axioms, proven theorems and so on) it must be on the other side (by implications or, sometimes, equivalence).
Consider $B = (B\cap A) \cup (B\cap A^c)$. Let's take an element $x\in (B\cap A) \cup (B\cap A^c)$.
$$ x \in (B\cap A) \cup (B\cap A^c) \iff x\in B\cap A \lor x\in B\cap A^c \; (\text{Definition of } \cup)$$
$$ \iff (x\in B \land x\in A)\lor (x\in B\land x\in A^c) \; (\text{Definition of } \cap) $$
Now, from this point on, do you know the definition of $A^c$?, do you know the distributive law between $\land$ and $\lor$?
A: Hint: In showing $B = (B \cap A) \cup (B \cap A^c)$, think of $(B \cap A)$ as "The part of $B$ contained in $A$" and think of  $(B \cap A^c)$ as "the part of $B$ outside of $A$." Convince yourself that the union of the two must be $B$. A Venn Diagram can help tremendously. As for showing this rigorously, show $B \subset  (B \cap A) \cup (B \cap A^c)$ and then show $(B \cap A) \cup (B \cap A^c) \subset B$ to establish equality of the two.
In showing $A \cup B = A \cup (B \cap A^c)$ simply apply some set algebra to $A \cup (B \cap A^c)$. Use the fact that for sets $X,Y,Z$, we know $X \cup (Y \cap Z) = (X \cup Y) \cap (X \cup Z)$.
