Truth or lie? "At least one [of the two] of us is a liar." 
This problem involves two people. Person A and person B. They can
  either always tell the truth, or always lie. When asked, person A
  replies that: "At least one of us is a liar." Does person A and B
  tell the truth or do they lie?

What I have so far is that person A can either tell the truth or lie. If he tells the truth, then his statement can be expressed by "A$\lor$B", if he lies then it must be given that they are both speaking the truth, which contradicts that he would be a liar. So person A must be speaking the truth.
But if person A is speaking the truth, then person B could be either lying or speaking the truth. Both would still make person A tell the truth.
So how can I figure out whether person B is speaking the truth or not?
 A: $$\begin{array}{cccc}
&&&\text{A says}\\
\text{A lies} &\text{B lies}&\text{A or B lies}&\text{A or B lies}\\
a&b&a\lor b&(a\lor b)\oplus a\\
F& F& F& \color{red}F\\
T& F& T& \color{red}F\\
F& T& T& \color{green}T\\
T& T& T& \color{red}F\\
\end{array}$$
A: If B would speak the truth then both of them would speak the truth - a contradiction to A's statement that at least one of them is a liar. So B will obviously not speak the truth.
A: I find it easier to always suppose that A, B tell the truth in any of these riddles with two types of people (that either always tells truth, or always lies). This is my approach:
$a$: A tells the truth
$b$: B tells the truth
Proposition "at least one of us is a liar" is $\lnot a \lor \lnot b$. Person A tells us this proposition, so $a$ and $\lnot a \lor \lnot b$ must have the same truth value (if A tells the truth, his proposition is true and if he lies, his proposition is false). Therefore, the answer to the problem is if there is one or more $T$ values of the proposition $a\leftrightarrow (\lnot a \lor \lnot b$). Let's see:
\begin{array}{cccccc}
a& b& \color{grey}{\lnot a}& \color{grey}{\lnot b}& \lnot a \lor \lnot b& a\leftrightarrow (\lnot a \lor \lnot b)\\
F& F& \color{grey}T& \color{grey}T& T& \color{red}F\\
T& F& \color{grey}F& \color{grey}T& T& \color{green}T\\
F& T& \color{grey}T& \color{grey}F& T& \color{red}F\\
T& T& \color{grey}F& \color{grey}F& F& \color{red}F\\
\end{array}
So, we see that $a\leftrightarrow (\lnot a \lor \lnot b)$ is true only when A tells the truth and B tells lies.
The same approach can be applied to any of this kind of problems.
