I am not entirely sure if I am going on the right pass here in solving the following logical puzzle:
There are 3 people.
$X$ says only one person is lying. $Y$ says exactly two people are lying. $Z$ says all of us are lying.
This is how I tried to solve it, I used the negation for lying and normal form for truth.
$Z: \lnot X\wedge \lnot Y\wedge \lnot Z$ so,
$\lnot Z: X \vee Y\vee Z$ , which forms a contradiction as Z can't be a truth teller
So, $Y: (\lnot X\wedge\lnot Y\wedge Z)\vee(\lnot X\wedge\lnot Z\wedge Y)\vee(\lnot Z\wedge\lnot Y\wedge X)$
But the first brackets form a contradiction as Z is a liar, so that can be canceled out
so,$Y:(\lnot X\wedge\lnot Z\wedge Y)\vee(\lnot Z\wedge\lnot Y\wedge X)$ Here I think I can't go any further because, both could be correct.
Thus, I move to the X,
$X:(\lnot X\wedge Y\wedge Z)\vee(\lnot Y\wedge Z\wedge X)\vee(\lnot Z\wedge Y\wedge X)$ Here again we cancel out two of them for the same reason like $Y$.
so, $X:(\lnot Z\wedge Y\wedge X)$
And here I just tried if $X$ is true then, $Y$ and $x$ are true which shows contradiction in $Y$ so $X$ is a liar as well and thus $Y$ is the truth teller?
Now, my question is whether this was correct, and if anyone can show me an alternative shorter/faster way to solve this? Thanks in advance.