My question is about that image:
Is this the correct Venn diagram for part c ? Why is there a red region in the middle that is painted ? Where did this red region came from?
And about the relation:
$\endgroup$ 9This is true?How can I visualize this using the Venn diagram? $$(A \Delta B) \Delta C = A \Delta (B \Delta C)$$
2 Answers
$\begingroup$You can see here that symmetric difference has an important property: The symmetric difference is associative! There are different ways to prove that, you can see in this related Math.SE questions: here and here.
And you can see other proof that I personally like here.
But your question is exactly:
Is this the correct diagram for the symmetric difference of the given A,B,C sets?
The answer is yes. There is a lot of ways to see it.
You can use the diagram to see what would be the symmetric difference:
$$A\Delta B$$
So the symmetric difference is the area of $A$ and $B$ without the intersection. See that the area taken out were the area of $A \cup B - (A\cap B)$. This is the definition of symmetric difference.
$$A\cap B$$
We have that the set $C$ is
$$C$$
Now, for the set $A\Delta B \Delta C$ we must take the areas of the both sets above and take the intersection. See that the center (in your question the red area) is in $C$ but is not in $A\Delta B$ so it is not an intersection, so should be considered! But the pink areas that are shown in the figure below are the intersection. So we will have
$$(A\Delta B) \cap C$$
So we take the area of all the unions and take out the pink areas. We are left with the symmetric difference of $A,B$ and $C$:
$$A\Delta B \Delta C $$
Other way is to use the relations here and construct a Venn diagram using the relations. For example
$$A\Delta B \Delta C = \displaystyle \left({A \cap \overline B \cap \overline C}\right) \cup \left({\overline A \cap B \cap \overline C}\right) \cup \left({\overline A \cap \overline B \cap C}\right) \cup \left({A \cap B \cap C}\right)$$
Where the bar above means that we take the complement of the set. If we name each of the sets that are been united: $X := \left({A \cap \overline B \cap \overline C}\right)$, $Y := \left({\overline A \cap B \cap \overline C}\right)$, $Z := \left({\overline A \cap \overline B \cap C}\right)$, $W := \left({A \cap B \cap C}\right)$ in this nomenclature we have $$A\Delta B\Delta C = X \cup Y \cup Z \cup W$$ We will have the above union and each of the sets will have the name that we give above inside them
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Here is another way of looking at this.
Logically, the symmetric difference corresponds to XOR:
$A \Delta B = \{ x|x \in A \: XOR \: x \in B \}$
Since the XOR is associative:
$A \: XOR \: (B \: XOR \: C) \Leftrightarrow (A \: XOR \: B) \: XOR \: C$
we can have XOR's with any number of terms.
Now, what would make a generalized XOR statement with $n$ terms true? It turns out that it is true when an odd number of terms are true.
Applied back to sets, this means that $A \Delta B \Delta C$ contains all elements that are in exactly an odd number of sets, i.e. in exactly one or all three. And that you see nicely in the Venn diagram.
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