How would you prove that
(A/B) U B = A if and only if B C A ?
There is no one way to do any set theory proof.
Here you might note that $\displaystyle \left( {A\backslash B} \right) \equiv \left( {A \cap B^c } \right)$
Starting with $\displaystyle B\subseteq A$ what would $\displaystyle \left( {A \cap B^c } \right)\cup A=~?$
I was thinking of relabelling (B C A) as D and ((A/B) U B = A) as E and then trying to argue D implies E and also that ~D implies ~E, but not sure if this would work or even where to start.
There is no one way to do any set theory proof.
Here you might note that $\displaystyle \left( {A\backslash B} \right) \equiv \left( {A \cap B^c } \right)$
Starting with $\displaystyle B\subseteq A$ what would $\displaystyle \left( {A \cap B^c } \right)\cup A=~?$
Should not $\displaystyle \left( {A \cap B^c } \right)\cup A$ be $\displaystyle \left( {A \cap B^c } \right)\cup B$
And then be equal to: $\displaystyle (B\cup A)\cap(B\cup B^c)$ ?