Set Theory Proof: Proving Identity (A U B) ∩ (B U C) ∩ (C U A)

[hmph]

Hi, I have been trying for a very long time to prove the following set theory identity

(A union B) intersect (B union C) intersect (C union A) = (A intersect B) union (B intersect C) union (C intersect A).

I thought that I could simplify (A U B) intersect (B U C) intersect (C U A)
as [B U (A intersect C) ]intersect (C U A), but venn diagrams show that this is not correct.

Thanks

 

[haruspex]

Do you know how to distribute union over intersection and vice versa?
(AuB)^(BuC) = (A^(BuC))u(B^(BuC))
Since B^(BuC) = B you can simplify that a little.
Just keep applying those rules and the result should drop out.
If you get stuck, post your working so far.

 

[hmph]

Thank you, it worked. I probably did it in a more convoluted manner than required, but I was able to do it in around 12 lines.

I didn't realize that you could distribute like that at first.