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## Homework Statement

Prove that for all sets A, B, and C, (A-C) [tex]\cap[/tex] (B-C) [tex]\cap[/tex] (A-B) = ∅

## Homework Equations

## The Attempt at a Solution

Proof: Suppose A, B, and C are sets

Let x [tex]\in[/tex] (A-C) [tex]\cap[/tex] (B-C) [tex]\cap[/tex] (A-B)

Since x [tex]\in[/tex] (A-C), by definition of difference, x [tex]\in[/tex] A and x [tex]\notin[/tex] C

Since x [tex]\in[/tex] (B-C), x [tex]\in[/tex] B and x [tex]\notin[/tex] C

Since x [tex]\in[/tex] (A-B), x [tex]\in[/tex] A and x [tex]\notin[/tex] B

Then by definition of intersection, if x [tex]\in[/tex] A then x [tex]\notin[/tex] C and x [tex]\notin[/tex] B

Also, if x [tex]\in[/tex] B then x [tex]\notin[/tex] C

Therefore there is no intersection of sets A, B, and C

Therefore, the intersection of (A-C) [tex]\cap[/tex] (B-C) [tex]\cap[/tex] (A-B) = ∅

Is this proof correct, I feel like I am missing something?