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?
$$\begin{array}{cccc}
&amp;&amp;&amp;\text{A says}\\
\text{A lies} &amp;\text{B lies}&amp;\text{A or B lies}&amp;\text{A or B lies}\\
a&amp;b&amp;a\lor b&amp;(a\lor b)\oplus a\\
F&amp; F&amp; F&amp; \color{red}F\\
T&amp; F&amp; T&amp; \color{red}F\\
F&amp; T&amp; T&amp; \color{green}T\\
T&amp; T&amp; T&amp; \color{red}F\\
\end{array}$$
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&amp; b&amp; \color{grey}{\lnot a}&amp; \color{grey}{\lnot b}&amp; \lnot a \lor \lnot b&amp; a\leftrightarrow (\lnot a \lor \lnot b)\\
F&amp; F&amp; \color{grey}T&amp; \color{grey}T&amp; T&amp; \color{red}F\\
T&amp; F&amp; \color{grey}F&amp; \color{grey}T&amp; T&amp; \color{green}T\\
F&amp; T&amp; \color{grey}T&amp; \color{grey}F&amp; T&amp; \color{red}F\\
T&amp; T&amp; \color{grey}F&amp; \color{grey}F&amp; F&amp; \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.