Representing the empty set on an Euler diagram

Question

I try to describe with Euler diagram the following relations: $A \oplus A = \emptyset$ and $\emptyset \subset (A \cap B) \subset (A \cup B)$. But empty set confuses me.

I even cannot imagine how to use Euler diagram with the first expression. But the second one I understand partially (I hope so):

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How in fact to show $\emptyset$ in Euler diagram like in expressions above?

What does $A \oplus A$ mean? Also, in the notation you're using, if $X \subset Y$, can $X = Y$? — Michael Albanese, Oct 21 '13 at 19:49
Yes, but in this case how one should show $\emptyset$ in Euler diagram? — Dragon, Oct 21 '13 at 19:55
Could you answer the second question from my first comment? That will help me to help you. — Michael Albanese, Oct 21 '13 at 19:56
There's no precise definition to it. But I think that one can consider X = Y in X $\subset$ Y. It is like X $\subset$ U, I think - X can be a part of Universal set or Universal set itself by content. — Dragon, Oct 21 '13 at 19:59
It is the same. I was asking about the symbol $\subset$. Some people write $X \subset Y$ to mean that $X$ is a subset of $Y$ but not equal to $Y$, whereas others allow the possibility that $X = Y$. It seems that you use the notation to allow equality between the two sets. — Michael Albanese, Oct 21 '13 at 20:08

score 1 · Accepted Answer · answered Oct 21 '13 at 20:52

The condition $A\oplus A = \emptyset$, where $\oplus$ is symmetric difference, is satisfied by every set $A$ as $$A\oplus A = (A\setminus A)\cup(A\setminus A) = \emptyset\cup\emptyset = \emptyset.$$ Therefore, if you want to draw an Euler diagram to represent this condition, just draw any set $A$.

The condition $\emptyset \subset A\cap B \subset A\cup B$, where $\subset$ allows for the possibility of equality, is satisfied by any two sets $A$ and $B$. Clearly $\emptyset \subset A\cap B$ as $\emptyset$ is a subset of every set, and any element which is in both $A$ and $B$ is in at least one of them so $A\cap B \subset A\cup B$. Therefore, to draw a Euler diagram to represent this condition, just draw any two sets $A$ and $B$.

Representing the empty set on an Euler diagram

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