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The problem is as follows:

First find the truth values of $p$, $q$ and $r$ such as the complex statement from below is false.

$$\lnot p \rightarrow (q\lor \lnot r)$$

Then using this information find the truth values of each of the following statements,

I. $\lnot (p \lor q) \rightarrow (p \leftrightarrow \not q)$

II. $(r \lor \lnot p) \bigtriangleup r$

III. $(\lnot p \bigtriangleup r)\lor (\lnot p \rightarrow q)$

Assuming you answer correctly all the statements I, II and III which would be your answer?.

The answers given in my book are as follows:

$\begin{array}{ll} 1.&\textrm{FFF}\\ 2.&\textrm{FFT}\\ 3.&\textrm{TFF}\\ 4.&\textrm{TFT}\\ \end{array}$

My book defines the $\bigtriangleup$ operator as a strong disjunction as follows from these equivalences:

$p \bigtriangleup q \equiv \lnot (p \leftrightarrow q)$

$p \bigtriangleup q \equiv (\lnot p \leftrightarrow q)$

$p \bigtriangleup q \equiv (p \lor q) \lor \lnot (p \land q)$

$p \bigtriangleup T \equiv \lnot p$

From then on I don't know exactly how to "guess" the adequate values which would make the statement to pinpoint the answer for the each of the statement.

Can someone help me here?. I don't know exactly what to do. The only thing which I do recall is that the definition of the $\bigtriangleup$ for a strong disjunction is given by this truth table.

$\begin{array}{|c|c|c|} \hline p& q & p \bigtriangleup q \\ \hline T&T&F\\ \hline T&F&T\\ \hline F&T&T\\ \hline F&F&F\\ \hline \end{array}$

But I don't know if this would help into the solution. Can this be used to find the values for getting the truth values for each of the statements given?. Can someone help me?.

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  • $\begingroup$ Hint: $A \to B$ is false iff $A$ is true but $B$ is false. $\endgroup$
    – player3236
    Commented Dec 25, 2020 at 19:09
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    $\begingroup$ @player3236 Thanks for that hint. Yes I noted about that. $\endgroup$
    – Chris Steinbeck Bell
    Commented Dec 25, 2020 at 20:33
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    $\begingroup$ I'm undoing the previous edit because that rollback has worsened Chris's post without making it any more aligned to the current sole answer (which fails to address Chris's question due primarily to the Answerer's own fault of not properly/fully reading the original Question). The Answerer—instead of rolling back the OP's edit—ought to simply just expand their answer! $\endgroup$
    – ryang
    Commented Jun 24, 2022 at 0:59

1 Answer 1

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$$\lnot p \rightarrow (q\lor \lnot r)$$ is false if and only if $\lnot p$ is true, and $(q\lor \lnot r)$ is false.

If $\lnot p$ is true, then $\lnot \lnot p \equiv p$ is not true.

$q\lor \lnot r$ is false only when both $q$ is false and $\lnot r$ is false.

$(p, q, r) = (F, F, T)$.

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  • $\begingroup$ Note, I answered your title question, and arrived at the correct answer (2). You then go on to discuss something entirely different than your title question: Symmetric difference. $\endgroup$
    – amWhy
    Commented Dec 25, 2020 at 19:31
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    $\begingroup$ The answer is not correct, although the problem indicates to find the values of $p$, $q$ and $r$, what it intends to ask is to find the truth value of the I, II and III statements. Hence this is the reason why I mentioned about the strong disjunction operator. Can you take a look at your answer again please?. $\endgroup$
    – Chris Steinbeck Bell
    Commented Dec 25, 2020 at 20:30
  • $\begingroup$ To reflect the changes I've updated the question. Please take a look. $\endgroup$
    – Chris Steinbeck Bell
    Commented Dec 25, 2020 at 20:38
  • $\begingroup$ You should not post questions with a "moving target": first you say one thing, someone answers, then you as a completely different question. That is rude. $\endgroup$
    – amWhy
    Commented Dec 25, 2020 at 20:40
  • $\begingroup$ I'm sorry if what I did seemed that way but this was not my intention. Because I translated this from a source in a different language I did it in a rush, this caused some confusion. I'll be more cautious next time. $\endgroup$
    – Chris Steinbeck Bell
    Commented Dec 25, 2020 at 20:46

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