Question on exclusive or vs inclusive or











up vote
1
down vote

favorite












Here's what my textbook asks me to prove:




From algebra, recall that, if a prime $p$ divides a product $ab$, then
$p$ must divide either $a$ or $b$. Use this to prove that $sqrt{p}$
is irrational for any prime $p$.




I know the proof follows exactly like the proof of irrationality of $sqrt{2}$. But my question is, what if $b=a$? Then the statement would follow like "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" which is false.



I wonder if I am doing some mistake here because the following video https://youtu.be/uQ6KSt94jVY seems to prove the same statement. I think it should be an "inclusive or" there.






logic







share|cite|improve this question






















  • How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
    – F.Carette
    Nov 22 at 10:31










  • @F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
    – Ashish K
    Nov 22 at 10:34












  • I see no reason to think that it's an exclusive or.
    – Andreas Blass
    Nov 22 at 17:37















up vote
1
down vote

favorite












Here's what my textbook asks me to prove:




From algebra, recall that, if a prime $p$ divides a product $ab$, then
$p$ must divide either $a$ or $b$. Use this to prove that $sqrt{p}$
is irrational for any prime $p$.




I know the proof follows exactly like the proof of irrationality of $sqrt{2}$. But my question is, what if $b=a$? Then the statement would follow like "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" which is false.



I wonder if I am doing some mistake here because the following video https://youtu.be/uQ6KSt94jVY seems to prove the same statement. I think it should be an "inclusive or" there.






logic







share|cite|improve this question






















  • How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
    – F.Carette
    Nov 22 at 10:31










  • @F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
    – Ashish K
    Nov 22 at 10:34












  • I see no reason to think that it's an exclusive or.
    – Andreas Blass
    Nov 22 at 17:37













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Here's what my textbook asks me to prove:




From algebra, recall that, if a prime $p$ divides a product $ab$, then
$p$ must divide either $a$ or $b$. Use this to prove that $sqrt{p}$
is irrational for any prime $p$.




I know the proof follows exactly like the proof of irrationality of $sqrt{2}$. But my question is, what if $b=a$? Then the statement would follow like "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" which is false.



I wonder if I am doing some mistake here because the following video https://youtu.be/uQ6KSt94jVY seems to prove the same statement. I think it should be an "inclusive or" there.






logic







share|cite|improve this question













Here's what my textbook asks me to prove:




From algebra, recall that, if a prime $p$ divides a product $ab$, then
$p$ must divide either $a$ or $b$. Use this to prove that $sqrt{p}$
is irrational for any prime $p$.




I know the proof follows exactly like the proof of irrationality of $sqrt{2}$. But my question is, what if $b=a$? Then the statement would follow like "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" which is false.



I wonder if I am doing some mistake here because the following video https://youtu.be/uQ6KSt94jVY seems to prove the same statement. I think it should be an "inclusive or" there.






logic




logic






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 22 at 10:26









Ashish K

775513




775513












  • How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
    – F.Carette
    Nov 22 at 10:31










  • @F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
    – Ashish K
    Nov 22 at 10:34












  • I see no reason to think that it's an exclusive or.
    – Andreas Blass
    Nov 22 at 17:37


















  • How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
    – F.Carette
    Nov 22 at 10:31










  • @F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
    – Ashish K
    Nov 22 at 10:34












  • I see no reason to think that it's an exclusive or.
    – Andreas Blass
    Nov 22 at 17:37
















How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
– F.Carette
Nov 22 at 10:31




How is the statement false? $ p | a lor p | a equiv p |a $. And $p | a^2 equiv p |a$ (for p prime, of course)
– F.Carette
Nov 22 at 10:31












@F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
– Ashish K
Nov 22 at 10:34






@F.Carette it's an exclusive or.. "exclusive or" is true if exactly one of the statements are true. In this particular case, both would be true.
– Ashish K
Nov 22 at 10:34














I see no reason to think that it's an exclusive or.
– Andreas Blass
Nov 22 at 17:37




I see no reason to think that it's an exclusive or.
– Andreas Blass
Nov 22 at 17:37










3 Answers
3






active

oldest

votes

















up vote
0
down vote



accepted










Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.



Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.






share|cite|improve this answer




























    up vote
    0
    down vote













    "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.






    share|cite|improve this answer




























      up vote
      0
      down vote













      The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
      is not problematic.



      for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.






      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: 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%2f3008961%2fquestion-on-exclusive-or-vs-inclusive-or%23new-answer', 'question_page');
        }
        );

        Post as a guest















        Required, but never shown

























        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes








        up vote
        0
        down vote



        accepted










        Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.



        Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.






        share|cite|improve this answer

























          up vote
          0
          down vote



          accepted










          Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.



          Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.






          share|cite|improve this answer























            up vote
            0
            down vote



            accepted







            up vote
            0
            down vote



            accepted






            Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.



            Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.






            share|cite|improve this answer












            Usually, in mathematical and logical contexts, "or" has to be interpreted as an inclusive disjunction, unless it is indicated otherwise.



            Note that $3$ (a prime number) divides not only $27 cdot 9$, but also $27$ and $9$. Hence, it is clear that the correct interpretation of the "or" in the statement "if a prime $p$ divides a product $ab$, then $p$ must divide either $a$ or $b$" is the "inclusive or", even when you consider the case where $a neq b$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Nov 22 at 10:38









            Taroccoesbrocco

            4,92761838




            4,92761838






















                up vote
                0
                down vote













                "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.






                share|cite|improve this answer

























                  up vote
                  0
                  down vote













                  "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.






                  share|cite|improve this answer























                    up vote
                    0
                    down vote










                    up vote
                    0
                    down vote









                    "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.






                    share|cite|improve this answer












                    "if $p$ divides $a^2$ then either $p$ divides $a$ or $p$ divides $a$" is correct if and only if the or is an inclusive "or". Hence you are correct.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Nov 22 at 10:32









                    Vee Hua Zhi

                    776124




                    776124






















                        up vote
                        0
                        down vote













                        The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
                        is not problematic.



                        for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.






                        share|cite|improve this answer

























                          up vote
                          0
                          down vote













                          The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
                          is not problematic.



                          for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.






                          share|cite|improve this answer























                            up vote
                            0
                            down vote










                            up vote
                            0
                            down vote









                            The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
                            is not problematic.



                            for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.






                            share|cite|improve this answer












                            The statement " if $p$ divides $a^2$ then $p$ divides $a$ or $p$ divides $a$ "
                            is not problematic.



                            for example $7$ divides $35^2$ so $7$ divides $35$ or $7$ divides $35$ is a true statement. This OR is inclusive and as you know $Plor P equiv P$.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Nov 22 at 10:38









                            Mohammad Riazi-Kermani

                            40.4k41958




                            40.4k41958






























                                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%2f3008961%2fquestion-on-exclusive-or-vs-inclusive-or%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

                                Ellipse (mathématiques)

                                Image
                                Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
                                Read more

                                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

                                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