Sets symmetric difference, proving elements belonging to $A, B$ or $C$ The question is: 
Let $A, B, C$ be arbitrary quantities and consider the symmetrical difference between two quantities:
$$A △ B = (A - B) ∪ (B - A).$$
Prove that each element contained in the quantity $A △ B △ C$ lies in exactly 1 or 3 of the quantities $A, B, C$.
I don't know how to solve this problem. What I'm thinking is that I should describe the subsets, give them some letters, but that's about how far I've gotten. 

How do i continue and solve a problem like this?
 A: Good start.  Now figure out which elements are in $A\triangle B$ and from there which items are in $A\triangle B\triangle C$. 
For instance, $$A\triangle B=(A\setminus B)\cup(B\setminus A)=\{a,b,e\}\cup\{f,g\}=\{a,b,e,f,g\}$$
A: The Venn Diagram method is a good one to see what is going on. If you require a more algebraic method then one can answer as follows:-
Let $x$ lie in none of $A,B,C$. Then $x$ is in neither $A\Delta B$ nor $C$ so is not in  $A\Delta B\Delta C.$
Let $x$ lie in precisely one of $A,B,C$, say $A$. Then $x$ is in $A\Delta B$ but not $C$ and so $x$ is in  $A\Delta B\Delta C.$ 
Let $x$ lie in precisely two of $A,B,C$, say $A$ and $B$. Then $x$ is  in neither $A\Delta B$ nor $C$ and so $x$ is not in  $A\Delta B\Delta C.$ 
Let $x$ lie in all three of $A,B,C$. Then $x$ is not in $A\Delta B$ nor $C$ but is in $x$ and so is in  $A\Delta B\Delta C.$ 
Thus $x$ is in $A\Delta B\Delta C$ if and only if it is in an odd number of $A,B,C$.
A: Let 
- A mean " x belongs to A" 


*

*B mean " x belongs to B" 

*C mean "x belongs to C"
The membership table shows that the formula correspondind to the set defined can be true when 2 of the above sentences are true at the same time ( see line 3 and 5). 
Here I use  as definition of symmetric difference : x belongs to A OR to B and does not belong to both A and B. 

