Boolean functions and characters 
I'm trying to solve this, but I have no idea how to start! I'm not even sure what the question is really asking. Can anybody rephrase this and perhaps give me pointers/solutions for one of the parts?

"Character" most likely refers to a "characteristic function", i.e. a function $\chi_S$ designed with a specific set $S$ in mind, such that $\chi_S(x)=1$ if $x\in S$ and $\chi_S(x)=0$ if not. So, for $XOR(x_1,x_2)$, the set of tuples $x$ on which $XOR(x)=1$ is $\{(0,1), (1,0)\}$. Then let that set be $S$. 
Can you take it from here?
Edit: I realized that the argument $x$ could be a vector of more than 2 entries, so in general the set $S$ for the XOR function should consist of all vectors such that either $(x_1,x_2)=(1,0)$ or $(x_1,x_2)=(0,1)$, and the other entries are arbitrary. 
