# Thread: Prove or Disprove the Following:

1. ## Prove or Disprove the Following:

I'm a little rusty and need to get back into the swing of things after not seeing this stuff for a while so I need a little push. Hope someone can help!

Prove or Disprove:

for any three sets A,B,C,

(a) A∪(B−C)⊂(A∪B)−(A∪C) Is this disproven by the sets A={1,2}, B={3,4}, and C={1,2}?

(b) (A∪B)−(A∪C)⊂A∪(B−C)

For any four sets A,B,C,D prove or disprove:

(c) (A×B)∪(C×D)=(A∪C)×(B∪D)

(d)(A×B)∩(C×D)=(A∩C)×(B∩D)

2. You do understand that this is not a homework service?
Therefore, you need to show some effort so that we know how to help.
We will no simply work these for you.

3. Originally Posted by Plato
You do understand that this is not a homework service?
Therefore, you need to show some effort so that we know how to help.
We will no simply work these for you.
Yes I am aware of that, but this was a result of getting frustrated. Anyway, I'm not going to bother working through (a), (c), or (d) because i'm pretty sure I've figured those out. However I'm completely stuck on (b).

My Work so far: Let x$\displaystyle \in$(A$\displaystyle \cup$B)-(A$\displaystyle \cup$C). Therefore x$\displaystyle \in$(A$\displaystyle \cup$B) and x$\displaystyle \notin$(A$\displaystyle \cup$C). Thus x$\displaystyle \in$A or x$\displaystyle \in$B.
Now im stuck, and I'm not even sure if my work so far is correct. I know I have to show that x$\displaystyle \in$ A$\displaystyle \cup$(B-C) I just don't know how to get there.

4. If $\displaystyle x\in (A \cup B)$ then $\displaystyle x\in A$ or $\displaystyle x\in B$.

If $\displaystyle x\notin (A \cup C)$ then $\displaystyle x\notin A$ and $\displaystyle x\notin C$.

Therefore, if $\displaystyle x\in (A \cup B) - (A \cup C)$, then $\displaystyle x\in B$. Actually, since $\displaystyle x\notin C$, we even have $\displaystyle x\in (B-C)$. Which is even stronger than what you need to show.

Henrik