How do you prove the statement: $A^c=(A\cup B)^c\cup (B \setminus A)$ I've been stuck on this question for a while and the problem is basically I just don't know how to prove it, I tried converting it to $\vee$ and $\wedge$ symbols then I did some research and found mathematically it doesn't make sense to compare them, so I'm currently stuck on how to prove such a statement.
Question
Determine whether the following statement is true or false, if true prove it, if false, provide a counterexample.
$A^c=(A\cup B)^c\cup (B \setminus A)$
Working
$A^c=$ ~$A$, $(A \cup B)^c=($~$A$ $\wedge$ ~$B$), $(B\setminus A)=$ ~$A$ 
$\implies A^c=(A\cup B)^c\cup (B \setminus A)=$~$A =($~$A$ $\wedge$ ~$B)$ $\wedge$ ~$A$ $\rightarrow$ I then went on to show this was equal to ~$A$ $\wedge $ ~$B$ which doesn't show anything additionally it's mathematically incorrect from what I found.
Note
I tried finding a counterexample and couldn't find one, so I'm assuming it is true, and as I said that's where I'm stuck, I don't know how to prove it. ANY help would GREATLY be appreciated, thanks! :)

I like Venn diagrams for showing equality of sets. $A^c$ is the pink section in the left figure. $(A\cup B)^c$ is the darker green in the right figure, and $B-A$ is the lighter green in the right figure.

