1
$\begingroup$

$p \Rightarrow q$ is true

Which is the logic value of $(p \vee r) \Rightarrow (q \vee r)$ ?

I did at this point and I can not siplify it more:

$\neg (p \vee r) \vee (q \vee r)$

$(\neg p \wedge \neg r) \vee ( q \vee r)$

Could someone help me please and explain how should I simplify it?

|cite|improve this question
$\endgroup$
  • 1
    $\begingroup$ Hint: Either $r$ is true or it is false ... $\endgroup$ – Henning Makholm Oct 23 '16 at 16:15
  • $\begingroup$ Well, there is also a way to see this without assuming excluded middle ;) $\endgroup$ – Oles Wohnzimmer Oct 23 '16 at 16:20
  • $\begingroup$ because of associativity you can drop parentheses around $q \vee r$. And then combine the $r$ with the $\neg p \wedge \neg r$ $\endgroup$ – Bram28 Oct 23 '16 at 16:23
  • $\begingroup$ Thank you @HenningMakholm , but when can I suppose that a letter is true or false? $\endgroup$ – Francisca Sousa Oct 23 '16 at 16:37
  • 1
    $\begingroup$ @FranciscaSousa: In classical logic you can always assume that anything you can write down (such as, for example, $r$) is either true or false. $\endgroup$ – Henning Makholm Oct 23 '16 at 16:45
0
$\begingroup$

If truth value of $r$ is $T$, then $p\lor T\implies q\lor T\equiv (T\implies T)\equiv T$.

If truth value of $r$ is $F$, then $p\lor F\implies q\lor F\equiv (p\implies q)\equiv T$.

|cite|improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.