Let U denote the universe and A, B, and C be three subsets. Prove that if A u B = A^c u C, then B u C = U. (I'll rewrite it in comments)

Question

anonymous · Answer

If $$A \cup B = A^c \cup C,$$ then $$B \cup C = U$$

anonymous · Answer

i think you're missing a piece of info...

anonymous · Answer

are A, B & C disjoint?

anonymous · Answer

All it told me was that...that would help though, wouldn't it?
I think just assume theyre not.. Does that make it impossible though?

zarkon · Answer

you have all the information you need to do the problem

anonymous · Answer

So do you know how to do it?

zarkon · Answer

I'll get you started

let $x\in U$

then $x\in A$ or $x\in A^c$

assume $x\in A$

then $x\in A\cup B$

therefore $x\in A^c\cup C$

since $x\in A$ we know that $x\notin A^c$, therefore $x\in C$


now show what happens if we assume $x\in A^{c}$

anonymous · Answer

How do you know $$x \in A^c \cup C$$ from $$x \in A \cup B$$?

zarkon · Answer

you are given

$$A \cup B = A^c \cup C$$

anonymous · Answer

Oh, duh...thanks lol

anonymous · Answer

So now assume $$x \in A^c$$ therefore $$x \in B$$ since $$x \notin A^c.$$ So $$x \in A \cup B$$ 
I dont know if I'm on the right track...

anonymous · Answer

*since $$x \notin A$$

zarkon · Answer

Assume $x\in A^c$ then $x\in A^c\cup C$...

anonymous · Answer

Okay thank you. My teacher sucks lol