Proving a statement of the form $Prightarrow(lnot Qlor lnot R)$.












0














Not looking for a proof of my question, only an answer to my question below.



Question: I want to assume that $P$ is true for this proof, then if I assume that $Q$ is also true and conclude that $R$ is not true, does that prove this statement? Do I also have to prove that if $R$ is true then $Q$ is not? Thank you for the help.



For reference the logical form of the statement: Suppose $f:[a,b]rightarrowBbb{R}$ is not constant on $[a,b]$, then $f$ is not continuous or $f([a,b])⊄Bbb{Q}^c$.






real-analysis logic proof-theory







share|cite|improve this question
























  • You might be better off proving the contrapositive of the statement . . .
    – Shaun
    Nov 27 '18 at 2:13
















0














Not looking for a proof of my question, only an answer to my question below.



Question: I want to assume that $P$ is true for this proof, then if I assume that $Q$ is also true and conclude that $R$ is not true, does that prove this statement? Do I also have to prove that if $R$ is true then $Q$ is not? Thank you for the help.



For reference the logical form of the statement: Suppose $f:[a,b]rightarrowBbb{R}$ is not constant on $[a,b]$, then $f$ is not continuous or $f([a,b])⊄Bbb{Q}^c$.






real-analysis logic proof-theory







share|cite|improve this question
























  • You might be better off proving the contrapositive of the statement . . .
    – Shaun
    Nov 27 '18 at 2:13














0












0








0


1





Not looking for a proof of my question, only an answer to my question below.



Question: I want to assume that $P$ is true for this proof, then if I assume that $Q$ is also true and conclude that $R$ is not true, does that prove this statement? Do I also have to prove that if $R$ is true then $Q$ is not? Thank you for the help.



For reference the logical form of the statement: Suppose $f:[a,b]rightarrowBbb{R}$ is not constant on $[a,b]$, then $f$ is not continuous or $f([a,b])⊄Bbb{Q}^c$.






real-analysis logic proof-theory







share|cite|improve this question















Not looking for a proof of my question, only an answer to my question below.



Question: I want to assume that $P$ is true for this proof, then if I assume that $Q$ is also true and conclude that $R$ is not true, does that prove this statement? Do I also have to prove that if $R$ is true then $Q$ is not? Thank you for the help.



For reference the logical form of the statement: Suppose $f:[a,b]rightarrowBbb{R}$ is not constant on $[a,b]$, then $f$ is not continuous or $f([a,b])⊄Bbb{Q}^c$.






real-analysis logic proof-theory




real-analysis logic proof-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 27 '18 at 19:58









Shaun

8,760113680




8,760113680










asked Nov 27 '18 at 1:47









Albert Diaz

925




925












  • You might be better off proving the contrapositive of the statement . . .
    – Shaun
    Nov 27 '18 at 2:13


















  • You might be better off proving the contrapositive of the statement . . .
    – Shaun
    Nov 27 '18 at 2:13
















You might be better off proving the contrapositive of the statement . . .
– Shaun
Nov 27 '18 at 2:13




You might be better off proving the contrapositive of the statement . . .
– Shaun
Nov 27 '18 at 2:13










1 Answer
1






active

oldest

votes


















1














You're right.



This is because $lnot Qlorlnot R$ is equivalent to $Qto lnot R$. There is no need to prove $Rtolnot Q$ if you prove the latter (due to the symmetry of $lor$).



Also: $$begin{align}
Pto(lnot Qlorlnot R)&equiv (lnot P)lor((lnot Q)lor(lnot R))\ &equiv ((lnot P)lor(lnot Q))lor(lnot R)\ &equiv (lnot(Pland Q))lorlnot R \ &equiv (Pland Q)to lnot R.end{align}$$






share|cite|improve this answer



















  • 1




    Thank you, logical statements can be confusing at times for me but now I recall that symmetry
    – Albert Diaz
    Nov 27 '18 at 2:03










  • You're welcome, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:09










  • I've added to the answer, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:10











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3015212%2fproving-a-statement-of-the-form-p-rightarrow-lnot-q-lor-lnot-r%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1














You're right.



This is because $lnot Qlorlnot R$ is equivalent to $Qto lnot R$. There is no need to prove $Rtolnot Q$ if you prove the latter (due to the symmetry of $lor$).



Also: $$begin{align}
Pto(lnot Qlorlnot R)&equiv (lnot P)lor((lnot Q)lor(lnot R))\ &equiv ((lnot P)lor(lnot Q))lor(lnot R)\ &equiv (lnot(Pland Q))lorlnot R \ &equiv (Pland Q)to lnot R.end{align}$$






share|cite|improve this answer



















  • 1




    Thank you, logical statements can be confusing at times for me but now I recall that symmetry
    – Albert Diaz
    Nov 27 '18 at 2:03










  • You're welcome, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:09










  • I've added to the answer, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:10
















1














You're right.



This is because $lnot Qlorlnot R$ is equivalent to $Qto lnot R$. There is no need to prove $Rtolnot Q$ if you prove the latter (due to the symmetry of $lor$).



Also: $$begin{align}
Pto(lnot Qlorlnot R)&equiv (lnot P)lor((lnot Q)lor(lnot R))\ &equiv ((lnot P)lor(lnot Q))lor(lnot R)\ &equiv (lnot(Pland Q))lorlnot R \ &equiv (Pland Q)to lnot R.end{align}$$






share|cite|improve this answer



















  • 1




    Thank you, logical statements can be confusing at times for me but now I recall that symmetry
    – Albert Diaz
    Nov 27 '18 at 2:03










  • You're welcome, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:09










  • I've added to the answer, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:10














1












1








1






You're right.



This is because $lnot Qlorlnot R$ is equivalent to $Qto lnot R$. There is no need to prove $Rtolnot Q$ if you prove the latter (due to the symmetry of $lor$).



Also: $$begin{align}
Pto(lnot Qlorlnot R)&equiv (lnot P)lor((lnot Q)lor(lnot R))\ &equiv ((lnot P)lor(lnot Q))lor(lnot R)\ &equiv (lnot(Pland Q))lorlnot R \ &equiv (Pland Q)to lnot R.end{align}$$






share|cite|improve this answer














You're right.



This is because $lnot Qlorlnot R$ is equivalent to $Qto lnot R$. There is no need to prove $Rtolnot Q$ if you prove the latter (due to the symmetry of $lor$).



Also: $$begin{align}
Pto(lnot Qlorlnot R)&equiv (lnot P)lor((lnot Q)lor(lnot R))\ &equiv ((lnot P)lor(lnot Q))lor(lnot R)\ &equiv (lnot(Pland Q))lorlnot R \ &equiv (Pland Q)to lnot R.end{align}$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 27 '18 at 19:57

























answered Nov 27 '18 at 2:02









Shaun

8,760113680




8,760113680








  • 1




    Thank you, logical statements can be confusing at times for me but now I recall that symmetry
    – Albert Diaz
    Nov 27 '18 at 2:03










  • You're welcome, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:09










  • I've added to the answer, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:10














  • 1




    Thank you, logical statements can be confusing at times for me but now I recall that symmetry
    – Albert Diaz
    Nov 27 '18 at 2:03










  • You're welcome, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:09










  • I've added to the answer, @AlbertDiaz :)
    – Shaun
    Nov 27 '18 at 2:10








1




1




Thank you, logical statements can be confusing at times for me but now I recall that symmetry
– Albert Diaz
Nov 27 '18 at 2:03




Thank you, logical statements can be confusing at times for me but now I recall that symmetry
– Albert Diaz
Nov 27 '18 at 2:03












You're welcome, @AlbertDiaz :)
– Shaun
Nov 27 '18 at 2:09




You're welcome, @AlbertDiaz :)
– Shaun
Nov 27 '18 at 2:09












I've added to the answer, @AlbertDiaz :)
– Shaun
Nov 27 '18 at 2:10




I've added to the answer, @AlbertDiaz :)
– Shaun
Nov 27 '18 at 2:10


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3015212%2fproving-a-statement-of-the-form-p-rightarrow-lnot-q-lor-lnot-r%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Quarter-circle Tiles

Image
up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2
Read more

build a pushdown automaton that recognizes the reverse language of a given pushdown automaton?

Image
0 I think it's similiar to NFAs. I replace the finite states of the given automaton for start-states for my new automaton. I do it with $epsilon$ -transition from the start state to the actual finite states of the given automaton. The start state of the given automaton will be my accepting(finite) state of my new one. All transitions will be reversed of course to recognize the reversed word. But how do I have to change the stack of the pushdown automaton? That's tricky. formal-languages automata context-free-grammar share | cite | improve this question edited Nov 29 '18 at 14:20 joee
Read more

Mont Emei

Image
Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog
Read more