Let A, B, and C be sets. Prove the following claim, or give a counterexample:

(A - B) - C = (A - C) - (B - C)

2. ## Set Proofs

Hello thehollow89
Originally Posted by thehollow89
Let A, B, and C be sets. Prove the following claim, or give a counterexample:

(A - B) - C = (A - C) - (B - C)
$\displaystyle (A-C)-(B-C) = (A\cap C')\cap(B\cap C')'$

$\displaystyle = (A\cap C')\cap(B'\cup C)$ 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

$\displaystyle =(A-B)-C$

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

$\displaystyle =(A-B)-C$

Is A-B = A n B? When n is the upside down u sign.

4. Originally Posted by thehollow89
Is A-B = A n B? When n is the upside down u sign.
$\displaystyle A\backslash B$ means A intersect B complement.
That is, all of A that is not in B.

5. ## Set notation

Hello thehollow89
Originally Posted by thehollow89
Is A-B = A n B? When n is the upside down u sign.
$\displaystyle A-B$, 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).