I tried searching for this (easy) question (both on here and on google in general), so if it's already been asked I apologize. My question concerns #13a in section 1.4, Operations on Sets in Velleman's How to Prove It. You are asked to prove the following:
(a)$ (A \Delta B) \cup C = (A \cup C) \Delta (B \setminus C)$ (Where Δ represents symmetric difference)
This should be a simple problem, but I keep getting hung up. Here's my work as follows:
$$ \begin{eqnarray*} (A Δ B) \cup C &=& [(A\setminus B) \cup (B \setminus A)] \cup C \\ &=& [(A \cap B') \cup (B \cap A')] \cup C \\ &=& [(A \cap B') \cup C] \cup [(B \cap A') \cup C] \\ &=& [(A \cup C) \cap (B' \cup C)] \cup [(B \cup C) \cap (A' \cup C)] \\ &=& [(A \cup C) \cap (B \cap C')'] \cup [(B \cup C) \cap (A' \cup C)] \\ &=& [(A \cup C) \setminus (B \ C)] \cup [(B \cup C) \cap (A' \cup C)] \end{eqnarray*} $$
My problem seems to stem from the second portion. I have that the RHS should equal the following when expanded
$$[(A \cup C) \setminus (B \setminus C)] \cup [(B \setminus C) \setminus (A \cup C)]$$
The problem I'm having is I see no way to get the second term to look like [(B\C) \ (A U C)]. If any of my logic is flawed please say so - and if there's a certain law I'm forgetting that'd wrap this up (forgot about deMorgan's for part of the problem for example, woops!) I'd greatly appreciate it. Again, sorry if this has already been asked.
Also, one last question: what's the most frequently used notation to denote symmetric difference within mathematics? I tend to like using the more common notation, just out of preference.
Thank you!
