Thread: Prove these statements are equivalent

1. Prove these statements are equivalent

Hello all, I'm a newbie at sets and logic, and I would like some help, if possible.

Suppose A, B and C are sets. Prove these statements are equivalent:

A (symmetric difference) C (intersects) B (symmetric difference) C = empty set

A (intersects) B (is a subset of) C (is a subset of) A (union) B

A (symmetric difference) C (is a subset of) A (symmetric difference) B

I would like some guidance on how to go about starting this problem. I started out by writing out the definition for symmetric difference (i.e. A (symmetric difference) C is equivalent to A (union) C \ A (intersects) C.)

2. Suppose A, B, and C are sets. Prove the following statements are equivalent.

(A∆C)∩(B∆C)=∅
(A∩B)⊆C⊆(A∪B)
(A∆C)⊆(A∆B)

Re-formatted because my first post was a little confusing to read.

The only thing I've done is re-write the symmetric differences out (i.e. (A∆C) is equivalent to (A∪C)\(A∩C)). A hint in the right direction would be much appreciated.

3. Originally Posted by KLM
Suppose A, B, and C are sets. Prove the following statements are equivalent.

(A∆C)∩(B∆C)=∅
(A∩B)⊆C⊆(A∪B)
(A∆C)⊆(A∆B)

Re-formatted because my first post was a little confusing to read.

The only thing I've done is re-write the symmetric differences out (i.e. (A∆C) is equivalent to (A∪C)\(A∩C)). A hint in the right direction would be much appreciated.
I'll help with one.
It's hard to hint with these without just doing it for you. Show us some work and we'll guide you.