I don't understand the following aspect of the parity problem and if someone could please explain it to me, I would be grateful. In the given quantum circuit, the output f(x) is defined to be x.a = x1a1+x2a2+x3a3(mod 2), where a is a fixed |a1a2a3>. For example, if x=|101> and a=|100>, x.a = 1+0+0(mod 2) = 1. I hope I got till this much correct. My question is, what if x is something like α|000>+β|101>+γ|111>? Then how is f(x) defined? Any help will be appreciated. Thanks!