Exercise of implication

Question

I have extracted this problem from the book How to Solve It by Daniel Velleman. Could someone explain me the solution? The problem is to determine the validity or invalidity of the argument. I would especially like to know how the conclusion is false by applying reasoning, not so much with the truth table (which I honestly don't understand much) that Velleman provides and whose image I attach.

" The warning light will come on if and only if the pressure is too high and the relief valve is clogged. The relief valve is not clogged. Therefore, the warning light will come on if and only if the pressure is too high."

Here is the solution:

solution by Velleman

Answer 1

The statement "The warning light will come on if and only if the pressure is too high and the relief valve is clogged" requires both the pressure to be too high and the relief valve to be clogged. If one of the two is false (such as in our case, the "The relief valve is not clogged"), then the warning light cannot come on.

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In the truth table, he is saying exactly this although, there is no explanation which is why I will attempt to explain it here.

We use the same notation: Let $W$ stand for "The warning light will come on," $P$ for "The pressure is too high," and $C$ for "The relief valve to be clogged."

We are given "The warning light will come on if and only if the pressure is too high and the relief valve is clogged," which translates to $W \iff (P \land C)$. We are also given "The relief valve is not clogged," which means $C$ is false. Then from the above statement, we know that $W$ is false. We are not given any information about $P$. In the line he presents, he considers the case when $P$ is true (because he wants to present a counterexample for which only one case is sufficient). From this information, it is obvious that $W \iff P$ is not true because $W$ is false in this case but $P$ is true.