Distributive/Associative Law Q

If I am trying to prove this:

For all sets A,B,C

(A intersection B)\C = (A\C) intersection (B\C)

*If I start with proving X is a subset of Y (L.S. = R.S.), could I do the following?*

(xeA ^ xeB) ^ (x/e/C)

Then using distributive law...

(xeA ^ x/e/C) ^ (xeB ^ x/e/C) <=> (A\C) intersection (B\C)

And then for Y is a subset of X...

Use the same distributive rule (backwards)

I'm wondering if this is a justified proof? If not, what would be the correct approach?

Thanks

Re: Distributive/Associative Law Q

Quote:

Originally Posted by

**DirectorRico** If I am trying to prove this:

For all sets A,B,C

(A intersection B)\C = (A\C) intersection (B\C)

$\displaystyle \begin{align*}(A\cap B)\setminus C&= (A\cap B)\cap C^c\\ &=(A\cap C^c)\cap (B\cap C^c) \\ &=(A\setminus C)\cap(B \setminus C) \end{align*}$.

Re: Distributive/Associative Law Q

Great! Thanks Plato. +rep