Clarification on the definition of truth

Question

So I am learning about the Godel's theorem. My instructor define truth and falsity as something arbitrary. he define $f$ to be true is $f(x)$ is $1$ and if $f(x) = 0$, it is false. I still dont have a good grasp on the theorem. He gave us an example. consider four $f$. $f(10) = 1$ and $f(10) = 0$, not $(f(10) = 1)$ and not $(f(10) = 0)$. Would the pair $f(10) = 0$ and not $(f(10) = 1)$ and the pair $f(10) = 1$ and not $(f(10) = 0)$ be true or all of them are true? (the range for $f$ is $\{0$ and $1\}$ and the domain of $f$ is all positive integer)