How to write the negation of a biconditional? I have two statements $p$ and $q$ such that:
Statement $p$: Ravi reads Mathematics
Statement $q$: Ravi reads Chemistry
Now, I am required to write the negation of $p \iff q$, i.e., I need to write $\lnot (p \iff q)$, which, if I understand correctly, is, by definition: $(p \land \lnot  q) \lor(\lnot p \land q )$
So, according to me, it should be as follows:
Ravi reads Mathematics and not Chemistry or Ravi doesn't read Mathematics and reads Chemistry.
But, the book states the answer to be as follows:
Ravi reads neither Mathematics nor Chemistry
Are the two statements equivalent? Or am I going wrong here?

Check question 3 (e) for the question asked here

The answer for question 3 (e)
You are exactly right, and your book is wrong.
Some equivalent ways to write $\lnot (p \iff q)$ are
$$
\begin{split}
\lnot (p \iff q) &amp;\equiv \lnot(p \implies q) \lor \lnot(q \implies p) \\
&amp;\equiv (p \land \lnot q) \lor (\lnot p \land q) \\
&amp;\equiv p \oplus q.
\end{split}
$$
(The last expression uses the XOR (&quot;exclusive or&quot;) operator, $\oplus$.)
However you choose to write it, it is equivalent to what you said:
&quot;Ravi reads Mathematics and not Chemistry or Ravi doesn't read Mathematics and reads Chemistry.
&quot;
OP, don't be discouraged by the rude or confusing answers in this thread. Trust in your logic!
A biconditional is implication that goes both ways. An implication is false exactly when $p$ is true and $q$ is false. Therefore it does not imply $q$. Additionally. The reverse implication could also be false, that $q$ is true and $p$ is false. Either one will work and both aren't required to render the biconditional false.
Simpler is that Ravis does not read both. I think you maybe are confused about the logical variables, and possibilities of true or false, they can take on, and then the variables as well as logical connectives.
edit the negation of the logical formula is one thing, but in an actual application of it, you would also have to negate the atomic sentences too, I think?
The author may be making a direct or possible indirect example of semantic truth vs syntatic truth, but we don't know because we don't know exactly what your book said.
We got a result in the book and so consider how it can be true, is literally what you've asked, not how to negate formula of biconditional.
$p,q=1$. That is what the semantics of a language might mean ultimately. So then the negation is clearly $p,q=0$. It's either a book typo or semantic truth.