# Is there a name for this set identity?

I proved that the following identity is true. It seems relatively simple and somewhat applicable, so I was wondering if there is a name for it (maybe something like De Morgan's Laws, for example).

$$(A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B)$$.

• There isn't a name for every set identity... – MathematicsStudent1122 Feb 23 '17 at 5:36
• @MathematicsStudent1122 I am aware, but it just seemed possible to me that this one might have a name (and if so, I would like to know what it is). – pzp Feb 23 '17 at 5:38
• The wiki page on the symmetric difference gives no name. en.wikipedia.org/wiki/Symmetric_difference – Thomas Andrews Feb 23 '17 at 6:00
• I don't think the identity itself has a name, but it is describing two equivalent definitions of the symmetric difference of two sets. – Dan Simon Feb 23 '17 at 6:02
• Well, I was going to suggest we call it pzp's law.... – fleablood Feb 24 '17 at 0:31

Definition: symmetric difference $$A \bigtriangleup B := (A \setminus B) \cup (B\setminus A)$$ Proof: $(A \cup B) \setminus (A \cap B)=(A \setminus B) \cup (B\setminus A)$ \begin{align*} (A \cup B)\setminus (A\cap B) &= ((A \cup B) \setminus A) \cup ((A \cup B) \setminus B)\\ &=(B \cup A) \setminus A) \cup ((A \cup B) \setminus B) \\ &= (B \setminus A) \cup (A \setminus B) \\ &=(A \setminus B) \cup (B\setminus A) \end{align*}