Set Proof Example I'm tasked to prove or disprove the following statement: $S \setminus (T \setminus R) = (S \setminus T) \setminus R$ (an element can be in multiple sets). I understand through visualizing with Venn Diagrams that this statement is false, but I'm having trouble simplifying the set minus operator for a proof.
 A: As someone else mentioned, it's enough to provide a counter example. Let $S=\{2,4\}, T=\{2,3\}$ and $R=\{1,2\}$. Then $(T\setminus R)=\{3\}$ and $(S\setminus T)=\{4\}$. Then $(S\setminus(T\setminus R))=\{2,4\}\neq ((S\setminus T)\setminus R)=\{4\}$
A: Disproof:
Let S={1,2,3,4,5} and T={1,2,3} and R={1,2}, we check for both sides of "equality":
For the L.H.S: $$ (S\setminus T) \setminus R = \left\lbrace 1,2,3,4,5\right\rbrace\setminus\left( \left\lbrace 1,2,3\right\rbrace \setminus \left\lbrace 1,2\right\rbrace\right) =  \left\lbrace 1,2,3,4,5\right\rbrace\setminus \left\lbrace 3\right\rbrace = \left\lbrace 1,2,4,5\right\rbrace $$
For the R.H.S:
$ \left(S\setminus T \right) \setminus R =  \left(\left\lbrace 1,2,3,4,5\right\rbrace\setminus \left\lbrace 1,2,3\right\rbrace\right)\setminus\left\lbrace 1,2\right\rbrace=\left\lbrace 4,5\right\rbrace \setminus\left\lbrace 1,2\right\rbrace =\left\lbrace 4,5\right\rbrace  $
Therefore,  $ S\setminus\left(T\setminus R\right) \ne \left(S\setminus T \right) \setminus R $  in general.
