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I know that the symmetric difference looks like that when having three sets A,B,C:

enter image description here

I want to make a prove with Venn diagrams to prove the associativity of the symmetric difference. What are some good ways to create this proof in a clear way?

I really appreciate your answer!

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marked as duplicate by Asaf Karagila, Dan Rust, martini, user1729, Hagen von Eitzen Jul 26 '13 at 13:13

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Writing "symmetric difference is associative" in the search box above gives this as a first result, and this as a close second. Does this question show any research efforts? – Asaf Karagila Jul 26 '13 at 12:32
up vote 5 down vote accepted

Essentially, to prove associativity of the symmetric difference of three sets, you are aiming to show that $$A \Delta (B \Delta C) = (A\Delta B)\Delta C\tag{1}$$

where, given any two sets, $X, Y$, $$X \Delta Y = (X \cup Y)\setminus (X \cap Y)$$

To show the equality given by $(1)$ using Venn Diagrams, you need only create a Venn Diagram corresponding to the set defined by the left-hand of $(1)$, and create a Venn Diagram that corresponds to the set defined by the right-hand side of $(1)$, and show that the two diagrams depict precisely the same set (i.e., that the two Venn diagrams are precisely the same).

It is also a worthwhile exercise to use, e.g., "element chasing" to provide an "algebraic" proof that the equality given by $(1)$ holds, and hence, that the symmetric difference is associative.

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