Validating Set Equivalencies: A Logical Approach

[Sorgen]

Homework Statement

Alright so I was trying to solve this using logical equivalences:

Fill in the blanks to make true identities:
$C \backslash ( A \Delta B) = (A \cap C) \Delta$ ______

I made it to the end where I stated that the missing part was (C\B), but I'm not sure if my last step was justified

Homework Equations

Equivalences 3. The Attempt at a Solution [\b]

I'll skip most of the steps (there were about 9) because I suck at latex but the last few are (working from the left side):

$[ ( x \in C \wedge x \in A) \wedge (x \notin A \vee x \in B) ] \vee [ (x \in C \wedge x \notin B) \wedge (x \notin A \vee x \in B) ] \\ [ (C \cap A) \cap (B \cup (x \notin A) ] \cap [ (C \backslash B) \cap (B \cup (x \notin A) ] \\ C \backslash (A \Delta B) = (A \cap C) \Delta (C \backslash B)$

So in the 2nd to last step, I dropped $$(B \cup (x \notin A)$$ from both sides of the union because of the definition of symmetric difference which says that they would be dropped even if I left them in. Is this correctly justified?

 

[Sorgen]

Screw latex, here's a scan of my work

anyway, in the 2nd to last step I dropped (B union ...) out of both sides due to the fact that they would get dropped anyway when symmetric difference was thrown in, is this justified?

 

[Sorgen]

Okay so I rewrote $[ ( B \cup ( x \notin A ) ]$ because I realized that notation doesn't make any sense (which I originally knew but wasn't sure how to express the -xEA part in set notation) and came up with $[ B \cup ( C \backslash A ) ]$ which I believe is a way of representing that in this context (where the only values we are talking about are represented in sets A B and C)

Is this rewrite correct or is there some way of representing -xEA using set notation that I don't know about?

Anyway, after plugging that in I realized that the 2nd to last line reads:
$[ ( C \cap A ) \cap ( B \cup ( C \backslash A ) ] \cup [ ( C \backslash B ) \cap ( B \cup ( C \backslash A ) ]$
And because of the first part which reads $[ ( C \cap A ) ]$, having $[ (C \cap A ) \cap ( C \backslash A ) ]$ is a contradiction, and because of that I can drop $( C \backslash A )$ from the equation on both sides.

Then, again because of the definition of symmetric difference (and because it's also a contradiction) I drop the $\cap B$ from both sides because it would be removed regardless.

Is this correct?