A phone rang in a class.
Four students each have their own phone.
We consider these four statements :
$P1$ : $\mathbf James$ : It's not my phone that rang.
$P2$ : $\mathbf Marie$ : It's James' phonee that rang Sir. \
$P3$ : $\mathbf Jacob$ : No Sir, It's Leyla's phone that rang.
$P4$ : $\mathbf Leyla$ answers : I confirm Sir that it's not James' phone that rang.
$P$ : Suppose only one student is saying the truth ( Only one proposition is true)
First question is : Justify $P1 \lor P4 = 0$.
I have one idea to justify this, but it just doesn't feel mathematical enough and its just intuitive.
What i thought of is that since $(P1 \Rightarrow P4) = 1 $ and $ (P4 \Rightarrow P1) = 1$, This can only be achieved if either both $P1$ and $P4$ are $True$ or both are $False$, Since they can't be both true then they're both $False$:
Hence $P1 \lor P4 = 0$
But it feels as if i'm not supposed to make the first two implications either or they're wrong.
Second question is : Justify $(P3 \Rightarrow P4) = 0$ Now the only case where an implication is $0$ is when $P3 = 1$ and we already know that $P4 = 0$, But how do i know that $P3$ is the true proposition?
Since $P1$ is $False$, then its negation is $True$ and isn't $P2 = \bar P1 = 1$ ?
and if $P3$ is $True$, doesn't that make both $P1$ and $P4$ true too because it's True that It isn't James' phone that rang?
I have trouble understanding the correct meaning in mathematical implication, it'd be really nice if someone can correct my way of interpreting it.