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Prove $A setminus (A setminus (A setminus B)) = A setminus B.$

Mathematics Asked by hk1510 on November 1, 2021

$renewcommand{backslash}{setminus}$

The question is as is in the title. I am able to show that $A setminus (A setminus (A setminus B)) subseteq A setminus B$ but I am stuck on showing that $A setminus B subseteq A setminus (A setminus (A setminus B))$.

Am I doing something wrong? Any suggestions will help.

Here is what I have so far:

$textbf{Proof:}$

To show that $A backslash (A backslash (A backslash B)) = A backslash B$, we need to show that $A backslash (A backslash (A backslash B)) subseteq A backslash B$ and $A backslash B subseteq A backslash (A backslash (A backslash B))$.

We first show that $A backslash (A backslash (A backslash B)) subseteq A backslash B$.

Let $a in A backslash (A backslash (A backslash B))$.

$implies a in A cap (A backslash (A backslash B))^c$

$implies a in A cap (A cap (A backslash B)^c)^c$

$implies a in A cap (A cap (A cap B^c)^c)^c$

$implies a in A cap (A^c cup (A cap B^c))$

$implies a in A cap ((A^c cup A) cap (A^c cup B^c))$

$implies a in A cap (U cap (A^c cup B^c))$

$implies a in A cap (A^c cup B^c)$

$implies a in (A cap A^c) cup (A cap B^c)$

$implies a in emptyset cup (A cap B^c)$

$implies a in (A cap B^c)$

$implies a in (A backslash B)$

Next, we show that $A backslash B subseteq A backslash (A backslash (A backslash B))$.

Let $a in A backslash B$.

$implies a in A backslash (A backslash B)^c$

$implies a in A backslash (A backslash (A backslash B)^c)^c$

$implies a in A backslash (A cap (A backslash B))^c$

$implies a in A backslash (A^c cup (A backslash B)^c)$

$implies a in A backslash (A^c cup (A^c cup B))$

2 Answers

Note that you can reverse the whole thing, i.e. if in the first half you simply change every $Rightarrow$ into a $Leftrightarrow$, you're there!

In fact, with algebra it's pretty simple:

$$A setminus (A setminus (A setminus B)) = $$

$$A cap (A cap (A cap B^c)^c)^c =$$

$$ A cap (A^c cup (A cap B^c) =$$

$$ (A cap A^c) cup (A cap A cap B^c) =$$

$$ emptyset cup (A cap B^c) =$$

$$ Asetminus B$$

Answered by Bram28 on November 1, 2021

Too long for a comment, but you could simplify notation quite a bit by writing $C = A setminus B$ and showing that $A setminus (A setminus C) = C$ for any set $C subset A$.

Note that if $D subset A$, then $x in D$ if and only if $x notin A setminus D$, and $x notin D$ if and only if $x in A setminus D$

So, let $C subset A$. Then $$x in C iff x notin A setminus C iff x in A setminus (A setminus C)$$ by applying the preceding remark to $D = C$ and $D = A setminus C$, respectively.

Answered by Umberto P. on November 1, 2021

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