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

[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.

 

[jbunniii]

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

Next, answer these questions:

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

and see what you can conclude.

 

[fauboca]

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.

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

 

[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).

 

[HallsofIvy]

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.

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

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

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

 

[Deveno]

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.

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).