# Thread: need to check one of De Morgan's proofs

1. ## need to check one of De Morgan's proofs

Hi

I have done the proof, just need to check it. I wanted to prove that

$A\cap B=A\setminus(A\setminus B)$

Proof:

$\mbox{let}\ x\in A\cap B\ \mbox{be arbitrary}$
$\Rightarrow x \in A\ \mbox{and}\ x \in B$
$\Rightarrow x \in A\ \mbox{and}\ x \notin B^{c}$
$\Rightarrow x \in A\ \mbox{and}\ x \notin A\setminus B$
$\Rightarrow x \in A\setminus (A\setminus B)$

I know that I have to do reverse proof too. But is it ok so far ?

Thanks

2. Yes. That is correct so far.

May I suggest that $A\setminus (A\setminus B)=A\cap(A\cap B^c)^c$.

3. are "complements" easier to manipulate that the "setminus" notation in proofs ?

4. Originally Posted by issacnewton
are "complements" easier to manipulate that the "setminus" notation in proofs ?
I think so. But each to his own.