Sets symmetric difference, proving elements belonging to $A, B$ or $C$

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, $$A\triangle B=(A\setminus B)\cup(B\setminus A)=\{a,b,e\}\cup\{f,g\}=\{a,b,e,f,g\}$$

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 $A,B,C$.