# 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?

• Discrete mathematics ??? – creative Sep 19 '14 at 8:59
• If at least one of $A,B$ is a liar and $A$ is not a liar, then ... – Hagen von Eitzen Sep 19 '14 at 9:05
• Jesus.. You are right Hagen... Been thinking too much about this I guess.. Thank you. – Attaque Sep 19 '14 at 9:12

## 3 Answers

$$\begin{array}{cccc} &&&\text{A says}\\ \text{A lies} &\text{B lies}&\text{A or B lies}&\text{A or B lies}\\ a&b&a\lor b&(a\lor b)\oplus a\\ F& F& F& \color{red}F\\ T& F& T& \color{red}F\\ F& T& T& \color{green}T\\ T& T& T& \color{red}F\\ \end{array}$$

• A or B lies is not $a \lor b$ but $\lnot a \lor \lnot b$ – Willemien Sep 19 '14 at 13:31
• @Willemien: no, $a$ (resp. $b$) denotes "A (B) lies"; the $\lor$ is non-exclusive. "A or B lies" is $a\lor b$. There is no doubt when you look at the truth values. The last column is the third one xored with "A lies", i.e. the truth value is changed when A lies. – Yves Daoust Sep 19 '14 at 13:48
• you are right sorry, I mis interpreted the $\oplus$ (Xor) (I do prefer the translation as the expressing $(\lnot a \lor \lnot b ) \equiv a$ so with using equivalences) – Willemien Sep 19 '14 at 13:49
• To my great surprise, this way of computing with logical expression makes it very straightforward. For instance "A says he lies" is $a\oplus a$, indeed a false statement. – Yves Daoust Sep 19 '14 at 13:57

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.

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.