# Prove that (A-B)-C=(A-C)-(B-C)

1. Feb 23, 2012

### iHeartof12

Let A,B and C be sets. Prove that
(A-B)-C=(A-C)-(B-C).

Attempted solution:

Suppose $x \in (A-B)-C$. Since $x \in (A-B)-C$ this means that $x \in A$ but $x \notin B$ and $x \notin C$.

I'm not sure how to show how these two statements are equal.

2. Feb 23, 2012

### jbunniii

So far so good. You have established that x is in A, but not in B and not in C.

* Is x in A - C?
* Is x in B - C?

and see what you can conclude.

3. Feb 23, 2012

### fauboca

Ok well you said $x\in A$ and $x\notin C$ What does that mean?

4. Feb 24, 2012

### iHeartof12

Let A,B and C be sets. Prove that
(A-B)-C=(A-C)-(B-C).

Attempted solution:

i.
Suppose $x \in (A-B)-C$. Since $x \in (A-B)-C$ this means that $x \in A$ but $x \notin B$ and $x \notin C$.

ii.
Suppose $x \in (A-C)-(B-C)$. Since $x \in (A-C)-(B-C)$ it makes since that $x \in A$ and $x \notin B$ and $x \notin C$.

Therefor these two statements are equal and (A-B)-C=(A-C)-(B-C).

5. Feb 24, 2012

### HallsofIvy

You need to finish this! $x \notin B$ and $x \notin C$ means what about x being in (A- C)- (B- C)?

Why dfoes that make sense? And what does that tell you about x being in (A- B)- C?

6. Feb 24, 2012

### Deveno

you want to show the two sets are subsets of each other; that is, that they have precisely the same elements.

if x is (A-B)-C, what does that mean?

first of all, it means x is in A-B, but x is not in C.

secondly, since x is in A-B, it means x is in A, but not in B.

putting these two statements together, we have: x is in A, x is not in B, x is not in C.

now if x is not in B, then it is not in B-C, since that is a subset of B.

(x is not only NOT in the part of B that lies outside of C, it's totally not in B anywhere).

but x IS in A, and x is NOT in C, so x IS in A-C.

so x IS in A-C and x is NOT in B-C, so x IS in (A-C)-(B-C).

that's "half" of the proof. the "other half" starts with assuming x is in (A-C)-(B-C).