Types of formula

Logical equivalence

Island of TuFa has two tribes, the Tu’s who always tell the truth, and the Fa’s who always lie.

A traveller encountered three resident of TuFa, $P$ , $Q$ , and $R$ , and each made a statement:

$P$ says ” $Q$ and $R$ tell the truth $⟺$ $R$ tells the truth”
$Q$ says “If $P$ and $R$ tell the truth, then it is not the case that if $Q$ and $R$ tell the truth, then $P$ tells the truth.”
$R$ says ” $Q$ is lying $⟺$ $P$ or $Q$ is telling the truth.”

Determine to which tribe each of $P$ , $Q$ , and $R$ belongs.

Let $p$ , $q$ and $r$ be whether $P$ , $Q$ and $R$ are telling the truth, respectively.

Formalising these statements:

We know that:

We can consider all of these in a truth table:

interpretation	$p$	$q$	$r$	$A$	$B$	$C$	$A \land B \land C$
$v_{0}$	0	0	0	0	0	1	0
$v_{1}$	0	0	1	1	0	0	0
$v_{2}$	0	1	0	0	1	1	0
$v_{3}$	0	1	1	0	1	0	0
$v_{4}$	1	0	0	1	0	0	0
$v_{5}$	1	0	1	0	1	1	0
$v_{6}$	1	1	0	1	1	1	1
$v_{7}$	1	1	1	1	0	0	0

$v_{6}$ is the only interpretation that makes the three statements, $A$ , $B$ , and $C$ true. Hence $P$ and $Q$ must belong to Tu, and $R$ to Fa.