How Can You Simplify the Set Expression (A ∪ B ∪ C) ∩ ((A ∩ B) ∪ C)?

[MartianVision]

Homework Statement

The question is, simplify this equation:
(A ∪ B ∪ C) ∩ ((A ∩ B) ∪ C)

The correct answer is (A ∩ B) ∪ C

Homework Equations

We have been given the commulative, associative, distributive, identity, complement and idempotent laws and DeMorgan's laws, and I researched the absorption laws myself.

The Attempt at a Solution

I tried doing:
(A ∪ B ∪ C) = x
(A ∩ B) = y
C = z <-- ((I know this isn't necessary, but thought it might make things easier)

From there I did:
(A ∪ B ∪ C) ∩ ((A ∩ B) ∪ C) = x ∩ (y ∪ z)
= (x ∪ y) ∩ (x ∪ z)
= ((A ∪ B ∪ C) ∪ (A ∩ B)) ∩ ((A ∪ B ∪ C) ∪ C)
= (A ∪ B ∪ C) ∩ (A ∪ B ∪ C) <-- I'm guessing I went wrong here, but I'm not too sure?
= (A ∪ B ∪ C)

I've tried a few other methods, but this one is the one that makes most sense.When I try to draw a venn diagram, I just don't understand how it isn't (A ∪ B ∪ C). I'm completely stumped :/

 

[phinds]

I hate those algebraic reductions, since K-maps are SO much easier (I'm an engineer ... I go for easy) and a K-map quickly shows that what you have listed as the correct answer IS the correct answer. I would think that a Venn diagram would show that fairly well also, but again, I prefer K-maps.

 

[MartianVision]

I just tried drawing another venn diagram, and I did get (A ∩ B) ∪ C. But the question is worded to find it out algebraically. So I'm a little reluctant to do just a Venn Diagram or K-Map to figure out the question, more just doing it using the algebraic laws of sets.

After drawing the Venn diagram, I'm right in thinking that the following logic is correct:
(These are made up equations, not out of the book)
if A ⊂ (B ∪ C)
then:
A ∩ (B ∪ C) = B ∪ C

If so, then to answer the question, the first step would be this:
(A ∪ B ∪ C) ∩ ((A ∩ B) ∪ C) = ((A ∩ B) ∪ C)