Let A, B, and C be sets. Prove the following claim, or give a counterexample:
(A - B) - C = (A - C) - (B - C)
Hello thehollow89 $\displaystyle (A-C)-(B-C) = (A\cap C')\cap(B\cap C')'$
$\displaystyle = (A\cap C')\cap(B'\cup C)$ De Morgan's Law
$\displaystyle = (A \cap C' \cap B')\cup(A \cap C' \cap C)$ Distributive Law
$\displaystyle = (A \cap B' \cap C')\cup(A \cap \oslash)$ Commutative Law, Complement Law
$\displaystyle = (A \cap B' \cap C')\cup \oslash$ Identity Law
$\displaystyle = (A \cap B') \cap C'$ Identity Law
$\displaystyle =(A-B)-C$
Grandad
Hello thehollow89$\displaystyle A-B$, also written (as Plato has done) $\displaystyle A\backslash B$, is the set of elements that are in $\displaystyle A$ and not in $\displaystyle B$; that is, the set $\displaystyle A \cap B'$ (notice the prime character $\displaystyle '$, after the $\displaystyle B$). You may find the notes at Discrete mathematics/Set theory - Wikibooks, collection of open-content textbooks helpful (which I have written under the name Nigeltn35).
Grandad