Working basic logic problems to solidify my grasp on the elementary concepts, I found the following kind of question a bit unclear and I am not sure whether my reasoning is appropriate to solve it.
An example of such a question is:
For which biconditional is its negation the following? $$(n^3\wedge (7n+2)\text{ are odd})\vee (n^3\wedge (7n+2)\text{ are even}) = P\vee Q$$ and is the biconditionnal true?
The way I understand the question is that the negation of some biconditional $\neg(P\iff Q)$ is equivalent to $P\vee Q$.
Thus, I think the question is "which biconditional's negation is the following disjunction" or "what is the biconditional that has its negation equivalent to this given disjunction".
The way I try to solve this is by realising that if $n$ is odd, then $$(n = 2k+1 :k\in \Bbb Z) \Rightarrow [(n^3 = 2(4k^3+6k^2+3k)+1)\wedge(7n+2=2(7k+4)+1)]$$ and if $n$ is even, then $$(n=2k:k\in \Bbb Z)\Rightarrow [(n^3=8k^3)\wedge(7n+2=2(7k+1)]$$ Hence, I think that if $P$ is $true$ it follows that $n$ is odd and $Q$ has to be $false$ and if $P$ is $false$ then $n$ is even and $Q$ has to be $true$.
So, I tried comparing the truth table of $P\vee Q$ with that of $\neg(P\iff Q)$ and what I see is that both have the same truth value when $P$ is $true$ and $Q$ is $false$ as well as when $P$ is $false$ and $Q$ is $true$.
In both cases, it follows that the biconditionnal is $false$.
I'm confused as to what to do from here because of the way the question is formulated in the book, it seems to ask for only one biconditional which makes me think that my whole reasoning is incorrect.
I would really appreciate getting some clarification on how to proceed with this kind of problem.
we have that $\neg (P \leftrightarrow Q) \Leftrightarrow (P \land \neg Q) \lor (\neg P \land Q)$
So, since the disjunction you end up with is
$(n^3 \: is \: odd \land 7n+2 \: is \: odd) \lor (n^3 \: is \: even \land 7n+2 \: is \: even)$,
the original biconditional must have been
$n^3 \: is \: odd \leftrightarrow 7n +2 \: is \: even$
And yes, that biconditional is obviously false: $n^3$ is odd iff $n$ is odd, while $7n+2$ is even iff $n$ is even.