Translate this sentence to predicate logic












1












$begingroup$


The question given asks to translate to predicate logic:




Every positive real number has a unique positive real root.




My solution to this problem is to separate it into the appropriate quantifiers.



C(x) = "Every positive real number x"



S(x) = "x has a positive real root"



the final logic form being: ∀x(C(x)→S(x))



It seems to easy for it to be correct. I need help in confirming my answer.






logic predicate-logic logic-translation







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    The question given asks to translate to predicate logic:




    Every positive real number has a unique positive real root.




    My solution to this problem is to separate it into the appropriate quantifiers.



    C(x) = "Every positive real number x"



    S(x) = "x has a positive real root"



    the final logic form being: ∀x(C(x)→S(x))



    It seems to easy for it to be correct. I need help in confirming my answer.






    logic predicate-logic logic-translation







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      The question given asks to translate to predicate logic:




      Every positive real number has a unique positive real root.




      My solution to this problem is to separate it into the appropriate quantifiers.



      C(x) = "Every positive real number x"



      S(x) = "x has a positive real root"



      the final logic form being: ∀x(C(x)→S(x))



      It seems to easy for it to be correct. I need help in confirming my answer.






      logic predicate-logic logic-translation







      share|cite|improve this question











      $endgroup$




      The question given asks to translate to predicate logic:




      Every positive real number has a unique positive real root.




      My solution to this problem is to separate it into the appropriate quantifiers.



      C(x) = "Every positive real number x"



      S(x) = "x has a positive real root"



      the final logic form being: ∀x(C(x)→S(x))



      It seems to easy for it to be correct. I need help in confirming my answer.






      logic predicate-logic logic-translation




      logic predicate-logic logic-translation






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 28 at 23:10









      Bram28

      63.9k44793




      63.9k44793










      asked Jan 26 at 22:32









      BotBot

      61




      61






















          2 Answers
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          active

          oldest

          votes


















          0












          $begingroup$

          The formula you have translates to:



          "For all $x$, if every positive real number $x$, then $x$ has a positive root".



          This does not make sense, and does not correspond to the statement you wish to formalize.



          What you want to say is




          $$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$




          i.e.,




          "For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.




          which is equivalent to




          "Every positive real number has a unique positive root."







          share|cite|improve this answer









          $endgroup$





















            0












            $begingroup$

            Couple of things:



            First, you don't want to define:




            $C(x)$ = "Every positive real number $x$"




            The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:




            $S(x)$ = "$x$ has a positive real root"




            The obvious thing to do is to define:



            $C(x)$ = "$x$ is a positive real number"



            Second, if we now look at



            $$forall x (C(x) rightarrow S(x))$$



            then we see that this means: "Every positive real number has a positive real root"



            .. but you were supposed to translate "Every positive real number has a unique positive real root"



            So, you're missing the "unique" part.



            What to do? One thing you can do is to simply redefine:



            $S(x)$ = "$x$ has a unique positive real root"



            But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$



            As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:



            $H(x,y)$: "$x$ has $y$ as a positive real root"



            OK, so now we can nicely re-express that "Every positive real number has a positive real root":



            $$forall x (C(x) rightarrow exists y H(x,y))$$



            Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:



            $$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$



            See how that works? There is a positive real root $y$, but there is not other possible real root $z$



            Another way to think about this is that every real root will have to be $y$. So:



            $$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$



            That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.



            Finally, though a little less intuitive, you can do:



            $$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$



            This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$






            share|cite|improve this answer









            $endgroup$













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              0












              $begingroup$

              The formula you have translates to:



              "For all $x$, if every positive real number $x$, then $x$ has a positive root".



              This does not make sense, and does not correspond to the statement you wish to formalize.



              What you want to say is




              $$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$




              i.e.,




              "For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.




              which is equivalent to




              "Every positive real number has a unique positive root."







              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                The formula you have translates to:



                "For all $x$, if every positive real number $x$, then $x$ has a positive root".



                This does not make sense, and does not correspond to the statement you wish to formalize.



                What you want to say is




                $$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$




                i.e.,




                "For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.




                which is equivalent to




                "Every positive real number has a unique positive root."







                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  The formula you have translates to:



                  "For all $x$, if every positive real number $x$, then $x$ has a positive root".



                  This does not make sense, and does not correspond to the statement you wish to formalize.



                  What you want to say is




                  $$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$




                  i.e.,




                  "For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.




                  which is equivalent to




                  "Every positive real number has a unique positive root."







                  share|cite|improve this answer









                  $endgroup$



                  The formula you have translates to:



                  "For all $x$, if every positive real number $x$, then $x$ has a positive root".



                  This does not make sense, and does not correspond to the statement you wish to formalize.



                  What you want to say is




                  $$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$




                  i.e.,




                  "For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.




                  which is equivalent to




                  "Every positive real number has a unique positive root."








                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 27 at 0:53









                  MetricMetric

                  1,22659




                  1,22659























                      0












                      $begingroup$

                      Couple of things:



                      First, you don't want to define:




                      $C(x)$ = "Every positive real number $x$"




                      The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:




                      $S(x)$ = "$x$ has a positive real root"




                      The obvious thing to do is to define:



                      $C(x)$ = "$x$ is a positive real number"



                      Second, if we now look at



                      $$forall x (C(x) rightarrow S(x))$$



                      then we see that this means: "Every positive real number has a positive real root"



                      .. but you were supposed to translate "Every positive real number has a unique positive real root"



                      So, you're missing the "unique" part.



                      What to do? One thing you can do is to simply redefine:



                      $S(x)$ = "$x$ has a unique positive real root"



                      But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$



                      As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:



                      $H(x,y)$: "$x$ has $y$ as a positive real root"



                      OK, so now we can nicely re-express that "Every positive real number has a positive real root":



                      $$forall x (C(x) rightarrow exists y H(x,y))$$



                      Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:



                      $$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$



                      See how that works? There is a positive real root $y$, but there is not other possible real root $z$



                      Another way to think about this is that every real root will have to be $y$. So:



                      $$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$



                      That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.



                      Finally, though a little less intuitive, you can do:



                      $$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$



                      This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$






                      share|cite|improve this answer









                      $endgroup$


















                        0












                        $begingroup$

                        Couple of things:



                        First, you don't want to define:




                        $C(x)$ = "Every positive real number $x$"




                        The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:




                        $S(x)$ = "$x$ has a positive real root"




                        The obvious thing to do is to define:



                        $C(x)$ = "$x$ is a positive real number"



                        Second, if we now look at



                        $$forall x (C(x) rightarrow S(x))$$



                        then we see that this means: "Every positive real number has a positive real root"



                        .. but you were supposed to translate "Every positive real number has a unique positive real root"



                        So, you're missing the "unique" part.



                        What to do? One thing you can do is to simply redefine:



                        $S(x)$ = "$x$ has a unique positive real root"



                        But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$



                        As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:



                        $H(x,y)$: "$x$ has $y$ as a positive real root"



                        OK, so now we can nicely re-express that "Every positive real number has a positive real root":



                        $$forall x (C(x) rightarrow exists y H(x,y))$$



                        Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:



                        $$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$



                        See how that works? There is a positive real root $y$, but there is not other possible real root $z$



                        Another way to think about this is that every real root will have to be $y$. So:



                        $$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$



                        That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.



                        Finally, though a little less intuitive, you can do:



                        $$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$



                        This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$






                        share|cite|improve this answer









                        $endgroup$
















                          0












                          0








                          0





                          $begingroup$

                          Couple of things:



                          First, you don't want to define:




                          $C(x)$ = "Every positive real number $x$"




                          The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:




                          $S(x)$ = "$x$ has a positive real root"




                          The obvious thing to do is to define:



                          $C(x)$ = "$x$ is a positive real number"



                          Second, if we now look at



                          $$forall x (C(x) rightarrow S(x))$$



                          then we see that this means: "Every positive real number has a positive real root"



                          .. but you were supposed to translate "Every positive real number has a unique positive real root"



                          So, you're missing the "unique" part.



                          What to do? One thing you can do is to simply redefine:



                          $S(x)$ = "$x$ has a unique positive real root"



                          But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$



                          As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:



                          $H(x,y)$: "$x$ has $y$ as a positive real root"



                          OK, so now we can nicely re-express that "Every positive real number has a positive real root":



                          $$forall x (C(x) rightarrow exists y H(x,y))$$



                          Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:



                          $$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$



                          See how that works? There is a positive real root $y$, but there is not other possible real root $z$



                          Another way to think about this is that every real root will have to be $y$. So:



                          $$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$



                          That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.



                          Finally, though a little less intuitive, you can do:



                          $$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$



                          This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$






                          share|cite|improve this answer









                          $endgroup$



                          Couple of things:



                          First, you don't want to define:




                          $C(x)$ = "Every positive real number $x$"




                          The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:




                          $S(x)$ = "$x$ has a positive real root"




                          The obvious thing to do is to define:



                          $C(x)$ = "$x$ is a positive real number"



                          Second, if we now look at



                          $$forall x (C(x) rightarrow S(x))$$



                          then we see that this means: "Every positive real number has a positive real root"



                          .. but you were supposed to translate "Every positive real number has a unique positive real root"



                          So, you're missing the "unique" part.



                          What to do? One thing you can do is to simply redefine:



                          $S(x)$ = "$x$ has a unique positive real root"



                          But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$



                          As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:



                          $H(x,y)$: "$x$ has $y$ as a positive real root"



                          OK, so now we can nicely re-express that "Every positive real number has a positive real root":



                          $$forall x (C(x) rightarrow exists y H(x,y))$$



                          Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:



                          $$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$



                          See how that works? There is a positive real root $y$, but there is not other possible real root $z$



                          Another way to think about this is that every real root will have to be $y$. So:



                          $$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$



                          That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.



                          Finally, though a little less intuitive, you can do:



                          $$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$



                          This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Jan 28 at 16:18









                          Bram28Bram28

                          63.9k44793




                          63.9k44793






























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