Can You Prove (A ∩ B) - C Equals (A - C) ∩ (B - C) in Set Theory?

[geforce]

Prove that (A n B) - C = (A - C) n (B - C).

n = intersect
≠ε = not a member

I got the first one by doing:
(xεA ^xεB) ^X≠εC ( by identity law and compliment law)


where would I go on from now?

 

[issacnewton]

so basically you have to prove the equality of sets. forward way is to let $x\in(A\cap B)\setminus C$ be arbitrary. Then as you have shown

$$x\in A\;x\in B\;\; x\notin C$$

which means that x is A and not in C AND x is in B and not in C. So just combine that to arrive at the right side. Then proceed to the reverse direction.

 

[geforce]

What do you mean as in "(A∩B)∖C"to be arbitrary and I have to show how it goes from (A∩B) -C to (A-C) ∩ (B-C)

 

[geforce]

?...

 

[issacnewton]

when you have to prove that two sets are equal you have to prove that

$$A\subseteq B\mbox{ and }B\subseteq A$$

So to prove $A\subseteq B$ you take arbitrary member of A and then prove that
its also member of B. And similar proof for proving $B\subseteq A$