Metaphorical Story about the irrefutability of the law of excluded middle

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
5
down vote

favorite












One can refute in intutionistic logic that they cannot refute the law of excluded middle. The proof is a bit strange:



em-irrefutable : ∀ A : Set → ¬ ¬ (A ⊎ ¬ A)
em-irrefutable k = k (inj₂ λ x → k (inj₁ x) )


In the discussion of this proof here, the author tells the following story:




Once upon a time, the devil approached a man and made an offer:
“Either (a) I will give you one billion dollars, or (b) I will grant
you any wish if you pay me one billion dollars. Of course, I get to
choose whether I offer (a) or (b).”



The man was wary. Did he need to sign over his soul? No, said the
devil, all the man need do is accept the offer.



The man pondered. If he was offered (b) it was unlikely that he would
ever be able to buy the wish, but what was the harm in having the
opportunity available?



“I accept,” said the man at last. “Do I get (a) or (b)?”



The devil paused. “I choose (b).”



The man was disappointed but not surprised. That was that, he thought.
But the offer gnawed at him. Imagine what he could do with his wish!
Many years passed, and the man began to accumulate money. To get the
money he sometimes did bad things, and dimly he realised that this
must be what the devil had in mind. Eventually he had his billion
dollars, and the devil appeared again.



“Here is a billion dollars,” said the man, handing over a valise
containing the money. “Grant me my wish!”



The devil took possession of the valise. Then he said, “Oh, did I say
(b) before? I’m so sorry. I meant (a). It is my great pleasure to give
you one billion dollars.”



And the devil handed back to the man the same valise that the man had
just handed to him.




In an attempt to interpret the proof in a nice way, I tried to write the proof as follows:



em-irrefutable : ∀ A : Set → ¬ ¬ (A ⊎ ¬ A)
em-irrefutable devil = devil (inj₂ λ billion → devil (inj₁ billion) )


But I do not quiet see how the story and the proof sync up, since what we should ideally be able to show in the proof is that it is possible to extract a billion dollars by tricking the devil. How do I better understand this proof by drawing a correspondence with the allegory?



I apologise if my question is too vague, but I find the proof to be a rather strange one, and would like to see an earthly explanation of it.




logic soft-question intuitionistic-logic popular-math




share|cite|improve this question



















  • I don't know about a precise answer but here's the way I see the allegory : proving $neg ( Alor neg A)$ is impossible because if you try to prove it to me, then I'll ask you to prove that neither $A$ nor $neg A$ holds, so you'll come to me all happy with a proof that $A$ does not hold (that's the billion dollars) and I'll say: well here you go, a proof of $neg A$. Anyway that's how the proof I know of this irrefutability goes, but I don't know if that's what the allegory is about
    – Max
    Jul 18 at 20:01










  • This seems to me to be a typical example of an "intuitive" or "motivational" explanation of a mathematical phenomenon that is much harder to internalise than the essence of the formal proof: the double negation mapping from a Heyting algebra to itself is a homomorphism.
    – Rob Arthan
    Jul 22 at 22:38














up vote
5
down vote

favorite












One can refute in intutionistic logic that they cannot refute the law of excluded middle. The proof is a bit strange:



em-irrefutable : ∀ A : Set → ¬ ¬ (A ⊎ ¬ A)
em-irrefutable k = k (inj₂ λ x → k (inj₁ x) )


In the discussion of this proof here, the author tells the following story:




Once upon a time, the devil approached a man and made an offer:
“Either (a) I will give you one billion dollars, or (b) I will grant
you any wish if you pay me one billion dollars. Of course, I get to
choose whether I offer (a) or (b).”



The man was wary. Did he need to sign over his soul? No, said the
devil, all the man need do is accept the offer.



The man pondered. If he was offered (b) it was unlikely that he would
ever be able to buy the wish, but what was the harm in having the
opportunity available?



“I accept,” said the man at last. “Do I get (a) or (b)?”



The devil paused. “I choose (b).”



The man was disappointed but not surprised. That was that, he thought.
But the offer gnawed at him. Imagine what he could do with his wish!
Many years passed, and the man began to accumulate money. To get the
money he sometimes did bad things, and dimly he realised that this
must be what the devil had in mind. Eventually he had his billion
dollars, and the devil appeared again.



“Here is a billion dollars,” said the man, handing over a valise
containing the money. “Grant me my wish!”



The devil took possession of the valise. Then he said, “Oh, did I say
(b) before? I’m so sorry. I meant (a). It is my great pleasure to give
you one billion dollars.”



And the devil handed back to the man the same valise that the man had
just handed to him.




In an attempt to interpret the proof in a nice way, I tried to write the proof as follows:



em-irrefutable : ∀ A : Set → ¬ ¬ (A ⊎ ¬ A)
em-irrefutable devil = devil (inj₂ λ billion → devil (inj₁ billion) )


But I do not quiet see how the story and the proof sync up, since what we should ideally be able to show in the proof is that it is possible to extract a billion dollars by tricking the devil. How do I better understand this proof by drawing a correspondence with the allegory?



I apologise if my question is too vague, but I find the proof to be a rather strange one, and would like to see an earthly explanation of it.




logic soft-question intuitionistic-logic popular-math




share|cite|improve this question



















  • I don't know about a precise answer but here's the way I see the allegory : proving $neg ( Alor neg A)$ is impossible because if you try to prove it to me, then I'll ask you to prove that neither $A$ nor $neg A$ holds, so you'll come to me all happy with a proof that $A$ does not hold (that's the billion dollars) and I'll say: well here you go, a proof of $neg A$. Anyway that's how the proof I know of this irrefutability goes, but I don't know if that's what the allegory is about
    – Max
    Jul 18 at 20:01










  • This seems to me to be a typical example of an "intuitive" or "motivational" explanation of a mathematical phenomenon that is much harder to internalise than the essence of the formal proof: the double negation mapping from a Heyting algebra to itself is a homomorphism.
    – Rob Arthan
    Jul 22 at 22:38












up vote
5
down vote

favorite









up vote
5
down vote

favorite











One can refute in intutionistic logic that they cannot refute the law of excluded middle. The proof is a bit strange:



em-irrefutable : ∀ A : Set → ¬ ¬ (A ⊎ ¬ A)
em-irrefutable k = k (inj₂ λ x → k (inj₁ x) )


In the discussion of this proof here, the author tells the following story:




Once upon a time, the devil approached a man and made an offer:
“Either (a) I will give you one billion dollars, or (b) I will grant
you any wish if you pay me one billion dollars. Of course, I get to
choose whether I offer (a) or (b).”



The man was wary. Did he need to sign over his soul? No, said the
devil, all the man need do is accept the offer.



The man pondered. If he was offered (b) it was unlikely that he would
ever be able to buy the wish, but what was the harm in having the
opportunity available?



“I accept,” said the man at last. “Do I get (a) or (b)?”



The devil paused. “I choose (b).”



The man was disappointed but not surprised. That was that, he thought.
But the offer gnawed at him. Imagine what he could do with his wish!
Many years passed, and the man began to accumulate money. To get the
money he sometimes did bad things, and dimly he realised that this
must be what the devil had in mind. Eventually he had his billion
dollars, and the devil appeared again.



“Here is a billion dollars,” said the man, handing over a valise
containing the money. “Grant me my wish!”



The devil took possession of the valise. Then he said, “Oh, did I say
(b) before? I’m so sorry. I meant (a). It is my great pleasure to give
you one billion dollars.”



And the devil handed back to the man the same valise that the man had
just handed to him.




In an attempt to interpret the proof in a nice way, I tried to write the proof as follows:



em-irrefutable : ∀ A : Set → ¬ ¬ (A ⊎ ¬ A)
em-irrefutable devil = devil (inj₂ λ billion → devil (inj₁ billion) )


But I do not quiet see how the story and the proof sync up, since what we should ideally be able to show in the proof is that it is possible to extract a billion dollars by tricking the devil. How do I better understand this proof by drawing a correspondence with the allegory?



I apologise if my question is too vague, but I find the proof to be a rather strange one, and would like to see an earthly explanation of it.




logic soft-question intuitionistic-logic popular-math




share|cite|improve this question











One can refute in intutionistic logic that they cannot refute the law of excluded middle. The proof is a bit strange:



em-irrefutable : ∀ A : Set → ¬ ¬ (A ⊎ ¬ A)
em-irrefutable k = k (inj₂ λ x → k (inj₁ x) )


In the discussion of this proof here, the author tells the following story:




Once upon a time, the devil approached a man and made an offer:
“Either (a) I will give you one billion dollars, or (b) I will grant
you any wish if you pay me one billion dollars. Of course, I get to
choose whether I offer (a) or (b).”



The man was wary. Did he need to sign over his soul? No, said the
devil, all the man need do is accept the offer.



The man pondered. If he was offered (b) it was unlikely that he would
ever be able to buy the wish, but what was the harm in having the
opportunity available?



“I accept,” said the man at last. “Do I get (a) or (b)?”



The devil paused. “I choose (b).”



The man was disappointed but not surprised. That was that, he thought.
But the offer gnawed at him. Imagine what he could do with his wish!
Many years passed, and the man began to accumulate money. To get the
money he sometimes did bad things, and dimly he realised that this
must be what the devil had in mind. Eventually he had his billion
dollars, and the devil appeared again.



“Here is a billion dollars,” said the man, handing over a valise
containing the money. “Grant me my wish!”



The devil took possession of the valise. Then he said, “Oh, did I say
(b) before? I’m so sorry. I meant (a). It is my great pleasure to give
you one billion dollars.”



And the devil handed back to the man the same valise that the man had
just handed to him.




In an attempt to interpret the proof in a nice way, I tried to write the proof as follows:



em-irrefutable : ∀ A : Set → ¬ ¬ (A ⊎ ¬ A)
em-irrefutable devil = devil (inj₂ λ billion → devil (inj₁ billion) )


But I do not quiet see how the story and the proof sync up, since what we should ideally be able to show in the proof is that it is possible to extract a billion dollars by tricking the devil. How do I better understand this proof by drawing a correspondence with the allegory?



I apologise if my question is too vague, but I find the proof to be a rather strange one, and would like to see an earthly explanation of it.





logic soft-question intuitionistic-logic popular-math





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 18 at 17:28









Agnishom Chattopadhyay

1,5441717




1,5441717











  • I don't know about a precise answer but here's the way I see the allegory : proving $neg ( Alor neg A)$ is impossible because if you try to prove it to me, then I'll ask you to prove that neither $A$ nor $neg A$ holds, so you'll come to me all happy with a proof that $A$ does not hold (that's the billion dollars) and I'll say: well here you go, a proof of $neg A$. Anyway that's how the proof I know of this irrefutability goes, but I don't know if that's what the allegory is about
    – Max
    Jul 18 at 20:01










  • This seems to me to be a typical example of an "intuitive" or "motivational" explanation of a mathematical phenomenon that is much harder to internalise than the essence of the formal proof: the double negation mapping from a Heyting algebra to itself is a homomorphism.
    – Rob Arthan
    Jul 22 at 22:38
















  • I don't know about a precise answer but here's the way I see the allegory : proving $neg ( Alor neg A)$ is impossible because if you try to prove it to me, then I'll ask you to prove that neither $A$ nor $neg A$ holds, so you'll come to me all happy with a proof that $A$ does not hold (that's the billion dollars) and I'll say: well here you go, a proof of $neg A$. Anyway that's how the proof I know of this irrefutability goes, but I don't know if that's what the allegory is about
    – Max
    Jul 18 at 20:01










  • This seems to me to be a typical example of an "intuitive" or "motivational" explanation of a mathematical phenomenon that is much harder to internalise than the essence of the formal proof: the double negation mapping from a Heyting algebra to itself is a homomorphism.
    – Rob Arthan
    Jul 22 at 22:38















I don't know about a precise answer but here's the way I see the allegory : proving $neg ( Alor neg A)$ is impossible because if you try to prove it to me, then I'll ask you to prove that neither $A$ nor $neg A$ holds, so you'll come to me all happy with a proof that $A$ does not hold (that's the billion dollars) and I'll say: well here you go, a proof of $neg A$. Anyway that's how the proof I know of this irrefutability goes, but I don't know if that's what the allegory is about
– Max
Jul 18 at 20:01




I don't know about a precise answer but here's the way I see the allegory : proving $neg ( Alor neg A)$ is impossible because if you try to prove it to me, then I'll ask you to prove that neither $A$ nor $neg A$ holds, so you'll come to me all happy with a proof that $A$ does not hold (that's the billion dollars) and I'll say: well here you go, a proof of $neg A$. Anyway that's how the proof I know of this irrefutability goes, but I don't know if that's what the allegory is about
– Max
Jul 18 at 20:01












This seems to me to be a typical example of an "intuitive" or "motivational" explanation of a mathematical phenomenon that is much harder to internalise than the essence of the formal proof: the double negation mapping from a Heyting algebra to itself is a homomorphism.
– Rob Arthan
Jul 22 at 22:38




This seems to me to be a typical example of an "intuitive" or "motivational" explanation of a mathematical phenomenon that is much harder to internalise than the essence of the formal proof: the double negation mapping from a Heyting algebra to itself is a homomorphism.
– Rob Arthan
Jul 22 at 22:38










1 Answer
1






active

oldest

votes

















up vote
2
down vote













It is useful to rewrite the proof using game semantics:
$$beginarrayl l c l l
textProponent & & & textOpponent & \
hline
1.; textasserts & negneg(A lor neg A) & & 2.; textattacks 1: & neg (A lor neg A) \
3.; textattacks 2: & A lor neg A & & 4.; textattacks 3: & ? \
5.; textdefends 3 from 4: & neg A & & 6.; textattacks 5: & A \
7.; textattacks 2: & A lor neg A & & 8.; textattacks 7: & ? \
9.; textdefends 7 from 8: & A & & - & \
hline
endarray$$



Different moments in the story correspond to different moves in the game, with the devil as the proponent and the man as the opponent.



Here $A$ can be interpreted as the proposition "The man has one billion dollars".
If the devil grants offer (a), it means that $A$ is true, because the man will have one billion dollars.
If the devil grants offer (b), it means that $neg A$ is true, because the offer has the form $A to B$ where $B$ is any wish, which is equivalent to $A to bot$ by the principle of explosion, which is exactly $neg A$ by definition.



When the devil makes his offer he's asserting the first proposition, $neg neg (A lor neg A)$, in move $1$. It sounds as though he's saying $A lor neg A$, but he must actually use the double negation in order to win (and it shouldn't be surprising that he's using deception, being the devil as he is).



The next part isn't reflected very well in the game, so we skip directly to move $4$, when the man (i.e., the opponent) asks if he gets offer (a) or (b). The devil responds with (b), which is move $5$ in the game.



Then the man gets his one billion dollars, which he presents to the devil with move $6$. But now the rules of the game allow the proponent to attack move $2$ again, which forces the opponent to counterattack, which in turn allows the proponent to win the game. This corresponds to the devil going back on his word (notice how move $9$ is the opposite of move $5$), choosing offer (a) and winning.






share|cite|improve this answer





















    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',
    convertImagesToLinks: true,
    noModals: false,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    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%2f2855812%2fmetaphorical-story-about-the-irrefutability-of-the-law-of-excluded-middle%23new-answer', 'question_page');

    );

    Post as a guest

















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    2
    down vote













    It is useful to rewrite the proof using game semantics:
    $$beginarrayl l c l l
    textProponent & & & textOpponent & \
    hline
    1.; textasserts & negneg(A lor neg A) & & 2.; textattacks 1: & neg (A lor neg A) \
    3.; textattacks 2: & A lor neg A & & 4.; textattacks 3: & ? \
    5.; textdefends 3 from 4: & neg A & & 6.; textattacks 5: & A \
    7.; textattacks 2: & A lor neg A & & 8.; textattacks 7: & ? \
    9.; textdefends 7 from 8: & A & & - & \
    hline
    endarray$$



    Different moments in the story correspond to different moves in the game, with the devil as the proponent and the man as the opponent.



    Here $A$ can be interpreted as the proposition "The man has one billion dollars".
    If the devil grants offer (a), it means that $A$ is true, because the man will have one billion dollars.
    If the devil grants offer (b), it means that $neg A$ is true, because the offer has the form $A to B$ where $B$ is any wish, which is equivalent to $A to bot$ by the principle of explosion, which is exactly $neg A$ by definition.



    When the devil makes his offer he's asserting the first proposition, $neg neg (A lor neg A)$, in move $1$. It sounds as though he's saying $A lor neg A$, but he must actually use the double negation in order to win (and it shouldn't be surprising that he's using deception, being the devil as he is).



    The next part isn't reflected very well in the game, so we skip directly to move $4$, when the man (i.e., the opponent) asks if he gets offer (a) or (b). The devil responds with (b), which is move $5$ in the game.



    Then the man gets his one billion dollars, which he presents to the devil with move $6$. But now the rules of the game allow the proponent to attack move $2$ again, which forces the opponent to counterattack, which in turn allows the proponent to win the game. This corresponds to the devil going back on his word (notice how move $9$ is the opposite of move $5$), choosing offer (a) and winning.






    share|cite|improve this answer

























      up vote
      2
      down vote













      It is useful to rewrite the proof using game semantics:
      $$beginarrayl l c l l
      textProponent & & & textOpponent & \
      hline
      1.; textasserts & negneg(A lor neg A) & & 2.; textattacks 1: & neg (A lor neg A) \
      3.; textattacks 2: & A lor neg A & & 4.; textattacks 3: & ? \
      5.; textdefends 3 from 4: & neg A & & 6.; textattacks 5: & A \
      7.; textattacks 2: & A lor neg A & & 8.; textattacks 7: & ? \
      9.; textdefends 7 from 8: & A & & - & \
      hline
      endarray$$



      Different moments in the story correspond to different moves in the game, with the devil as the proponent and the man as the opponent.



      Here $A$ can be interpreted as the proposition "The man has one billion dollars".
      If the devil grants offer (a), it means that $A$ is true, because the man will have one billion dollars.
      If the devil grants offer (b), it means that $neg A$ is true, because the offer has the form $A to B$ where $B$ is any wish, which is equivalent to $A to bot$ by the principle of explosion, which is exactly $neg A$ by definition.



      When the devil makes his offer he's asserting the first proposition, $neg neg (A lor neg A)$, in move $1$. It sounds as though he's saying $A lor neg A$, but he must actually use the double negation in order to win (and it shouldn't be surprising that he's using deception, being the devil as he is).



      The next part isn't reflected very well in the game, so we skip directly to move $4$, when the man (i.e., the opponent) asks if he gets offer (a) or (b). The devil responds with (b), which is move $5$ in the game.



      Then the man gets his one billion dollars, which he presents to the devil with move $6$. But now the rules of the game allow the proponent to attack move $2$ again, which forces the opponent to counterattack, which in turn allows the proponent to win the game. This corresponds to the devil going back on his word (notice how move $9$ is the opposite of move $5$), choosing offer (a) and winning.






      share|cite|improve this answer























        up vote
        2
        down vote










        up vote
        2
        down vote









        It is useful to rewrite the proof using game semantics:
        $$beginarrayl l c l l
        textProponent & & & textOpponent & \
        hline
        1.; textasserts & negneg(A lor neg A) & & 2.; textattacks 1: & neg (A lor neg A) \
        3.; textattacks 2: & A lor neg A & & 4.; textattacks 3: & ? \
        5.; textdefends 3 from 4: & neg A & & 6.; textattacks 5: & A \
        7.; textattacks 2: & A lor neg A & & 8.; textattacks 7: & ? \
        9.; textdefends 7 from 8: & A & & - & \
        hline
        endarray$$



        Different moments in the story correspond to different moves in the game, with the devil as the proponent and the man as the opponent.



        Here $A$ can be interpreted as the proposition "The man has one billion dollars".
        If the devil grants offer (a), it means that $A$ is true, because the man will have one billion dollars.
        If the devil grants offer (b), it means that $neg A$ is true, because the offer has the form $A to B$ where $B$ is any wish, which is equivalent to $A to bot$ by the principle of explosion, which is exactly $neg A$ by definition.



        When the devil makes his offer he's asserting the first proposition, $neg neg (A lor neg A)$, in move $1$. It sounds as though he's saying $A lor neg A$, but he must actually use the double negation in order to win (and it shouldn't be surprising that he's using deception, being the devil as he is).



        The next part isn't reflected very well in the game, so we skip directly to move $4$, when the man (i.e., the opponent) asks if he gets offer (a) or (b). The devil responds with (b), which is move $5$ in the game.



        Then the man gets his one billion dollars, which he presents to the devil with move $6$. But now the rules of the game allow the proponent to attack move $2$ again, which forces the opponent to counterattack, which in turn allows the proponent to win the game. This corresponds to the devil going back on his word (notice how move $9$ is the opposite of move $5$), choosing offer (a) and winning.






        share|cite|improve this answer













        It is useful to rewrite the proof using game semantics:
        $$beginarrayl l c l l
        textProponent & & & textOpponent & \
        hline
        1.; textasserts & negneg(A lor neg A) & & 2.; textattacks 1: & neg (A lor neg A) \
        3.; textattacks 2: & A lor neg A & & 4.; textattacks 3: & ? \
        5.; textdefends 3 from 4: & neg A & & 6.; textattacks 5: & A \
        7.; textattacks 2: & A lor neg A & & 8.; textattacks 7: & ? \
        9.; textdefends 7 from 8: & A & & - & \
        hline
        endarray$$



        Different moments in the story correspond to different moves in the game, with the devil as the proponent and the man as the opponent.



        Here $A$ can be interpreted as the proposition "The man has one billion dollars".
        If the devil grants offer (a), it means that $A$ is true, because the man will have one billion dollars.
        If the devil grants offer (b), it means that $neg A$ is true, because the offer has the form $A to B$ where $B$ is any wish, which is equivalent to $A to bot$ by the principle of explosion, which is exactly $neg A$ by definition.



        When the devil makes his offer he's asserting the first proposition, $neg neg (A lor neg A)$, in move $1$. It sounds as though he's saying $A lor neg A$, but he must actually use the double negation in order to win (and it shouldn't be surprising that he's using deception, being the devil as he is).



        The next part isn't reflected very well in the game, so we skip directly to move $4$, when the man (i.e., the opponent) asks if he gets offer (a) or (b). The devil responds with (b), which is move $5$ in the game.



        Then the man gets his one billion dollars, which he presents to the devil with move $6$. But now the rules of the game allow the proponent to attack move $2$ again, which forces the opponent to counterattack, which in turn allows the proponent to win the game. This corresponds to the devil going back on his word (notice how move $9$ is the opposite of move $5$), choosing offer (a) and winning.







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered Jul 18 at 22:44









        Luca Bressan

        3,86521036




        3,86521036






















             

            draft saved


            draft discarded


























             


            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2855812%2fmetaphorical-story-about-the-irrefutability-of-the-law-of-excluded-middle%23new-answer', 'question_page');

            );

            Post as a guest
































































            Comments

            Post a Comment

            Popular posts from this blog

            What is the equation of a 3D cone with generalised tilt?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
            Read more

            Color the edges and diagonals of a regular polygon

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
            Read more

            Relationship between determinant of matrix and determinant of adjoint?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
            Read more