# Thread: Set Theory equality proof

1. ## Set Theory equality proof

Hello,

I'm having difficulties with proving the following:
A \ (B\C) = (A\B) UNION (A INTERSECTION C)

I know it's correct, I can draw Venn diagrams that show it, but have no idea how to proof it using words (or other statements).

Any help will be appreciated.

Thanks!

2. ## Re: Set Theory equality proof

Originally Posted by BscStud
I'm having difficulties with proving the following:
A \ (B\C) = (A\B) UNION (A INTERSECTION C)

\displaystyle \begin{align*}A\setminus(B\setminus C) &=A\cap\overline{B\cap\overline{C}} \\&= A\cap (\overline{B}\cup C)\\&=(A\cap\overline{B})\cup(A\cap C)\end{align*}

Can you finish?

3. ## Re: Set Theory equality proof

Originally Posted by Plato
\displaystyle \begin{align*}A\setminus(B\setminus C) &=A\cap\overline{B\cap\overline{C}} \\&= A\cap (\overline{B}\cup C)\\&=(A\cap\overline{B})\cup(A\cap C)\end{align*}

Can you finish?
The concept is very clear to me, though only because I know negation from other study fields. In the course I'm undertaking now we did not study anything relating to negation of sets thus I'm afraid I cannot use it in profing the statement above (I know... University rules).

Is there any other way to proof it without negation? I do not need nor want a complete solution, just how to start.

Thank you very much for your respond.

4. ## Re: Set Theory equality proof

Originally Posted by BscStud
The concept is very clear to me, though only because I know negation from other study fields. In the course I'm undertaking now we did not study anything relating to negation of sets thus I'm afraid I cannot use it in profing the statement above (I know... University rules).

Is there any other way to proof it without negation? I do not need nor want a complete solution, just how to start.

Thank you very much for your respond.
That is totally absurd.

You can use the "pick-a-point" approach.
$\displaystyle t\in A\setminus(B\setminus C)\text{ iff }t\in A \text{ and }t\notin(B\setminus C)$

$\displaystyle t\notin(B\setminus C)\text{ iff }t\notin B\text{ or }t\in C$.

You then put those together.

5. ## Re: Set Theory equality proof

Trust me, I had a week long argue over this with my professor, mainly because I'm a long time computer programmer but they don't let us use any previous knowledge to solve basic tasks resulting in a lower-standards solutions.

This approach is what I was looking for, thanks a lot Plato. You've been a big help.