Determine whether f is a function, an injection, a surjection












2












$begingroup$


Let $P={p(x)$ | $p(x)$ is a polynomial of degree $n$, $n in Bbb Z^+cup{0} $ with coefficients in $Bbb R }$. Define $f : Prightarrow P$ where $f(p(x)) =p'(x)$, the derivative of $p(x)$.



Determine whether $f$ is a function, an injection, a surjection, a bijection.



Now I have the solutions, and I understand that it is a function because each polynomial has a unique derivative. And it is not an injection as the antiderivative of a given polynomial is not unique.



However, I do not understand the book's solution for determining whether it is a surjection, nor am I able to come up with one myself. In all honesty, I think I am having trouble understanding the mapping from P to P. The solution states that it is a surjection. Why is this so?










share|cite|improve this question









$endgroup$












  • $begingroup$
    This problem doesn't make sense. If $deg p(x)=n$, then $deg p'(x)=n-1$ and therefore, $p'(x)notin P$.
    $endgroup$
    – José Carlos Santos
    2 hours ago










  • $begingroup$
    Is $P$ the set of polynomials of degree $n$ exactly?
    $endgroup$
    – Bernard
    2 hours ago






  • 2




    $begingroup$
    I interpreted it to mean that it is the set of all polynomials.
    $endgroup$
    – Tony S.F.
    2 hours ago










  • $begingroup$
    I think he means the set of nonzero polynomials with coefficients in $mathbb R$.
    $endgroup$
    – Clayton
    2 hours ago






  • 1




    $begingroup$
    I read it more as the set of "all polynomials with degree that is a nonnegative integer." Though I do wonder of the need to specify that.
    $endgroup$
    – Eevee Trainer
    2 hours ago
















2












$begingroup$


Let $P={p(x)$ | $p(x)$ is a polynomial of degree $n$, $n in Bbb Z^+cup{0} $ with coefficients in $Bbb R }$. Define $f : Prightarrow P$ where $f(p(x)) =p'(x)$, the derivative of $p(x)$.



Determine whether $f$ is a function, an injection, a surjection, a bijection.



Now I have the solutions, and I understand that it is a function because each polynomial has a unique derivative. And it is not an injection as the antiderivative of a given polynomial is not unique.



However, I do not understand the book's solution for determining whether it is a surjection, nor am I able to come up with one myself. In all honesty, I think I am having trouble understanding the mapping from P to P. The solution states that it is a surjection. Why is this so?










share|cite|improve this question









$endgroup$












  • $begingroup$
    This problem doesn't make sense. If $deg p(x)=n$, then $deg p'(x)=n-1$ and therefore, $p'(x)notin P$.
    $endgroup$
    – José Carlos Santos
    2 hours ago










  • $begingroup$
    Is $P$ the set of polynomials of degree $n$ exactly?
    $endgroup$
    – Bernard
    2 hours ago






  • 2




    $begingroup$
    I interpreted it to mean that it is the set of all polynomials.
    $endgroup$
    – Tony S.F.
    2 hours ago










  • $begingroup$
    I think he means the set of nonzero polynomials with coefficients in $mathbb R$.
    $endgroup$
    – Clayton
    2 hours ago






  • 1




    $begingroup$
    I read it more as the set of "all polynomials with degree that is a nonnegative integer." Though I do wonder of the need to specify that.
    $endgroup$
    – Eevee Trainer
    2 hours ago














2












2








2


0



$begingroup$


Let $P={p(x)$ | $p(x)$ is a polynomial of degree $n$, $n in Bbb Z^+cup{0} $ with coefficients in $Bbb R }$. Define $f : Prightarrow P$ where $f(p(x)) =p'(x)$, the derivative of $p(x)$.



Determine whether $f$ is a function, an injection, a surjection, a bijection.



Now I have the solutions, and I understand that it is a function because each polynomial has a unique derivative. And it is not an injection as the antiderivative of a given polynomial is not unique.



However, I do not understand the book's solution for determining whether it is a surjection, nor am I able to come up with one myself. In all honesty, I think I am having trouble understanding the mapping from P to P. The solution states that it is a surjection. Why is this so?










share|cite|improve this question









$endgroup$




Let $P={p(x)$ | $p(x)$ is a polynomial of degree $n$, $n in Bbb Z^+cup{0} $ with coefficients in $Bbb R }$. Define $f : Prightarrow P$ where $f(p(x)) =p'(x)$, the derivative of $p(x)$.



Determine whether $f$ is a function, an injection, a surjection, a bijection.



Now I have the solutions, and I understand that it is a function because each polynomial has a unique derivative. And it is not an injection as the antiderivative of a given polynomial is not unique.



However, I do not understand the book's solution for determining whether it is a surjection, nor am I able to come up with one myself. In all honesty, I think I am having trouble understanding the mapping from P to P. The solution states that it is a surjection. Why is this so?







calculus functions derivatives elementary-set-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 hours ago









John ArgJohn Arg

496




496












  • $begingroup$
    This problem doesn't make sense. If $deg p(x)=n$, then $deg p'(x)=n-1$ and therefore, $p'(x)notin P$.
    $endgroup$
    – José Carlos Santos
    2 hours ago










  • $begingroup$
    Is $P$ the set of polynomials of degree $n$ exactly?
    $endgroup$
    – Bernard
    2 hours ago






  • 2




    $begingroup$
    I interpreted it to mean that it is the set of all polynomials.
    $endgroup$
    – Tony S.F.
    2 hours ago










  • $begingroup$
    I think he means the set of nonzero polynomials with coefficients in $mathbb R$.
    $endgroup$
    – Clayton
    2 hours ago






  • 1




    $begingroup$
    I read it more as the set of "all polynomials with degree that is a nonnegative integer." Though I do wonder of the need to specify that.
    $endgroup$
    – Eevee Trainer
    2 hours ago


















  • $begingroup$
    This problem doesn't make sense. If $deg p(x)=n$, then $deg p'(x)=n-1$ and therefore, $p'(x)notin P$.
    $endgroup$
    – José Carlos Santos
    2 hours ago










  • $begingroup$
    Is $P$ the set of polynomials of degree $n$ exactly?
    $endgroup$
    – Bernard
    2 hours ago






  • 2




    $begingroup$
    I interpreted it to mean that it is the set of all polynomials.
    $endgroup$
    – Tony S.F.
    2 hours ago










  • $begingroup$
    I think he means the set of nonzero polynomials with coefficients in $mathbb R$.
    $endgroup$
    – Clayton
    2 hours ago






  • 1




    $begingroup$
    I read it more as the set of "all polynomials with degree that is a nonnegative integer." Though I do wonder of the need to specify that.
    $endgroup$
    – Eevee Trainer
    2 hours ago
















$begingroup$
This problem doesn't make sense. If $deg p(x)=n$, then $deg p'(x)=n-1$ and therefore, $p'(x)notin P$.
$endgroup$
– José Carlos Santos
2 hours ago




$begingroup$
This problem doesn't make sense. If $deg p(x)=n$, then $deg p'(x)=n-1$ and therefore, $p'(x)notin P$.
$endgroup$
– José Carlos Santos
2 hours ago












$begingroup$
Is $P$ the set of polynomials of degree $n$ exactly?
$endgroup$
– Bernard
2 hours ago




$begingroup$
Is $P$ the set of polynomials of degree $n$ exactly?
$endgroup$
– Bernard
2 hours ago




2




2




$begingroup$
I interpreted it to mean that it is the set of all polynomials.
$endgroup$
– Tony S.F.
2 hours ago




$begingroup$
I interpreted it to mean that it is the set of all polynomials.
$endgroup$
– Tony S.F.
2 hours ago












$begingroup$
I think he means the set of nonzero polynomials with coefficients in $mathbb R$.
$endgroup$
– Clayton
2 hours ago




$begingroup$
I think he means the set of nonzero polynomials with coefficients in $mathbb R$.
$endgroup$
– Clayton
2 hours ago




1




1




$begingroup$
I read it more as the set of "all polynomials with degree that is a nonnegative integer." Though I do wonder of the need to specify that.
$endgroup$
– Eevee Trainer
2 hours ago




$begingroup$
I read it more as the set of "all polynomials with degree that is a nonnegative integer." Though I do wonder of the need to specify that.
$endgroup$
– Eevee Trainer
2 hours ago










3 Answers
3






active

oldest

votes


















6












$begingroup$

To see that $f$ is a surjection we take an arbitrary element $y$ in $P$ and show that $exists xin P$ such that $f(x)=y$. This is what it means to be surjective; we cover the entire space with the image of $f$ on $P$.



Let $pin P$, i.e. $p$ is some polynomial $p(x)$. Then $p(x)$ has an antiderivative, $q(x) = int p(x)dx$. This antiderivative is a polynomial (easy to check), so it is in $P$. Then $f(q) = p$, and since $p$ was arbitrary, $f$ is therefore surjective.



Here is a more concrete analogy to help you understand what a surjection is.



Imagine you have two lists of names, one for adults and one for children. Imagine that someone says they have a way to assign names from the adult list to names in children list. Their way of assigning the names is surjective if every name is the children list is assigned at least one name in the adult list. This would mean that if you pick a name from the children list as random, there is at least one adult name corresponding to it.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    This is exactly the solution presented in my book (albeit reworded) yet for some reason I am having trouble understanding it. It seems to be that I am having a problem with the first two lines. Perhaps I do not properly understand what surjection is?
    $endgroup$
    – John Arg
    2 hours ago



















1












$begingroup$

For any polynomial $p (x) $ there exists its integral $P (x) $, for which $p $ is a derivative: $$p (x)=P'(x)$$ and which itself is a real-coefficients polynomial, too $$P(x)in P$$ Hence each $p(x) in P$ is in the image of your function $f$. Then $f$ is surjective.






share|cite|improve this answer











$endgroup$





















    0












    $begingroup$

    Think about what the inverse mapping for $f$ would be if indeed it would be a surjection: it would take the derivatives to the original function. Phrased another way, it would take a function to its antiderivative, right?



    Well, we know that all polynomials have an antiderivative (which is itself a polynomial), and since the codomain of $f$ is basically the set of real polynomials, we have a pre-image for each polynomial in $P$ - its antiderivative. Thus, $f$ is surjective.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      What do you mean by pre-image?
      $endgroup$
      – John Arg
      2 hours ago










    • $begingroup$
      You know how, say, $f(x)$ might be called the "image" for $x$? Same sort of deal: we would also say $x$ is the pre-image for $f(x)$. To use, arguably, more familiar terminology: inputs are to outputs, as pre-images are to images.
      $endgroup$
      – Eevee Trainer
      2 hours ago












    Your Answer








    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%2f3187852%2fdetermine-whether-f-is-a-function-an-injection-a-surjection%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









    6












    $begingroup$

    To see that $f$ is a surjection we take an arbitrary element $y$ in $P$ and show that $exists xin P$ such that $f(x)=y$. This is what it means to be surjective; we cover the entire space with the image of $f$ on $P$.



    Let $pin P$, i.e. $p$ is some polynomial $p(x)$. Then $p(x)$ has an antiderivative, $q(x) = int p(x)dx$. This antiderivative is a polynomial (easy to check), so it is in $P$. Then $f(q) = p$, and since $p$ was arbitrary, $f$ is therefore surjective.



    Here is a more concrete analogy to help you understand what a surjection is.



    Imagine you have two lists of names, one for adults and one for children. Imagine that someone says they have a way to assign names from the adult list to names in children list. Their way of assigning the names is surjective if every name is the children list is assigned at least one name in the adult list. This would mean that if you pick a name from the children list as random, there is at least one adult name corresponding to it.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      This is exactly the solution presented in my book (albeit reworded) yet for some reason I am having trouble understanding it. It seems to be that I am having a problem with the first two lines. Perhaps I do not properly understand what surjection is?
      $endgroup$
      – John Arg
      2 hours ago
















    6












    $begingroup$

    To see that $f$ is a surjection we take an arbitrary element $y$ in $P$ and show that $exists xin P$ such that $f(x)=y$. This is what it means to be surjective; we cover the entire space with the image of $f$ on $P$.



    Let $pin P$, i.e. $p$ is some polynomial $p(x)$. Then $p(x)$ has an antiderivative, $q(x) = int p(x)dx$. This antiderivative is a polynomial (easy to check), so it is in $P$. Then $f(q) = p$, and since $p$ was arbitrary, $f$ is therefore surjective.



    Here is a more concrete analogy to help you understand what a surjection is.



    Imagine you have two lists of names, one for adults and one for children. Imagine that someone says they have a way to assign names from the adult list to names in children list. Their way of assigning the names is surjective if every name is the children list is assigned at least one name in the adult list. This would mean that if you pick a name from the children list as random, there is at least one adult name corresponding to it.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      This is exactly the solution presented in my book (albeit reworded) yet for some reason I am having trouble understanding it. It seems to be that I am having a problem with the first two lines. Perhaps I do not properly understand what surjection is?
      $endgroup$
      – John Arg
      2 hours ago














    6












    6








    6





    $begingroup$

    To see that $f$ is a surjection we take an arbitrary element $y$ in $P$ and show that $exists xin P$ such that $f(x)=y$. This is what it means to be surjective; we cover the entire space with the image of $f$ on $P$.



    Let $pin P$, i.e. $p$ is some polynomial $p(x)$. Then $p(x)$ has an antiderivative, $q(x) = int p(x)dx$. This antiderivative is a polynomial (easy to check), so it is in $P$. Then $f(q) = p$, and since $p$ was arbitrary, $f$ is therefore surjective.



    Here is a more concrete analogy to help you understand what a surjection is.



    Imagine you have two lists of names, one for adults and one for children. Imagine that someone says they have a way to assign names from the adult list to names in children list. Their way of assigning the names is surjective if every name is the children list is assigned at least one name in the adult list. This would mean that if you pick a name from the children list as random, there is at least one adult name corresponding to it.






    share|cite|improve this answer











    $endgroup$



    To see that $f$ is a surjection we take an arbitrary element $y$ in $P$ and show that $exists xin P$ such that $f(x)=y$. This is what it means to be surjective; we cover the entire space with the image of $f$ on $P$.



    Let $pin P$, i.e. $p$ is some polynomial $p(x)$. Then $p(x)$ has an antiderivative, $q(x) = int p(x)dx$. This antiderivative is a polynomial (easy to check), so it is in $P$. Then $f(q) = p$, and since $p$ was arbitrary, $f$ is therefore surjective.



    Here is a more concrete analogy to help you understand what a surjection is.



    Imagine you have two lists of names, one for adults and one for children. Imagine that someone says they have a way to assign names from the adult list to names in children list. Their way of assigning the names is surjective if every name is the children list is assigned at least one name in the adult list. This would mean that if you pick a name from the children list as random, there is at least one adult name corresponding to it.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited 2 hours ago

























    answered 2 hours ago









    Tony S.F.Tony S.F.

    3,72421031




    3,72421031












    • $begingroup$
      This is exactly the solution presented in my book (albeit reworded) yet for some reason I am having trouble understanding it. It seems to be that I am having a problem with the first two lines. Perhaps I do not properly understand what surjection is?
      $endgroup$
      – John Arg
      2 hours ago


















    • $begingroup$
      This is exactly the solution presented in my book (albeit reworded) yet for some reason I am having trouble understanding it. It seems to be that I am having a problem with the first two lines. Perhaps I do not properly understand what surjection is?
      $endgroup$
      – John Arg
      2 hours ago
















    $begingroup$
    This is exactly the solution presented in my book (albeit reworded) yet for some reason I am having trouble understanding it. It seems to be that I am having a problem with the first two lines. Perhaps I do not properly understand what surjection is?
    $endgroup$
    – John Arg
    2 hours ago




    $begingroup$
    This is exactly the solution presented in my book (albeit reworded) yet for some reason I am having trouble understanding it. It seems to be that I am having a problem with the first two lines. Perhaps I do not properly understand what surjection is?
    $endgroup$
    – John Arg
    2 hours ago











    1












    $begingroup$

    For any polynomial $p (x) $ there exists its integral $P (x) $, for which $p $ is a derivative: $$p (x)=P'(x)$$ and which itself is a real-coefficients polynomial, too $$P(x)in P$$ Hence each $p(x) in P$ is in the image of your function $f$. Then $f$ is surjective.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      For any polynomial $p (x) $ there exists its integral $P (x) $, for which $p $ is a derivative: $$p (x)=P'(x)$$ and which itself is a real-coefficients polynomial, too $$P(x)in P$$ Hence each $p(x) in P$ is in the image of your function $f$. Then $f$ is surjective.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        For any polynomial $p (x) $ there exists its integral $P (x) $, for which $p $ is a derivative: $$p (x)=P'(x)$$ and which itself is a real-coefficients polynomial, too $$P(x)in P$$ Hence each $p(x) in P$ is in the image of your function $f$. Then $f$ is surjective.






        share|cite|improve this answer











        $endgroup$



        For any polynomial $p (x) $ there exists its integral $P (x) $, for which $p $ is a derivative: $$p (x)=P'(x)$$ and which itself is a real-coefficients polynomial, too $$P(x)in P$$ Hence each $p(x) in P$ is in the image of your function $f$. Then $f$ is surjective.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 1 hour ago

























        answered 2 hours ago









        CiaPanCiaPan

        10.4k11248




        10.4k11248























            0












            $begingroup$

            Think about what the inverse mapping for $f$ would be if indeed it would be a surjection: it would take the derivatives to the original function. Phrased another way, it would take a function to its antiderivative, right?



            Well, we know that all polynomials have an antiderivative (which is itself a polynomial), and since the codomain of $f$ is basically the set of real polynomials, we have a pre-image for each polynomial in $P$ - its antiderivative. Thus, $f$ is surjective.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              What do you mean by pre-image?
              $endgroup$
              – John Arg
              2 hours ago










            • $begingroup$
              You know how, say, $f(x)$ might be called the "image" for $x$? Same sort of deal: we would also say $x$ is the pre-image for $f(x)$. To use, arguably, more familiar terminology: inputs are to outputs, as pre-images are to images.
              $endgroup$
              – Eevee Trainer
              2 hours ago
















            0












            $begingroup$

            Think about what the inverse mapping for $f$ would be if indeed it would be a surjection: it would take the derivatives to the original function. Phrased another way, it would take a function to its antiderivative, right?



            Well, we know that all polynomials have an antiderivative (which is itself a polynomial), and since the codomain of $f$ is basically the set of real polynomials, we have a pre-image for each polynomial in $P$ - its antiderivative. Thus, $f$ is surjective.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              What do you mean by pre-image?
              $endgroup$
              – John Arg
              2 hours ago










            • $begingroup$
              You know how, say, $f(x)$ might be called the "image" for $x$? Same sort of deal: we would also say $x$ is the pre-image for $f(x)$. To use, arguably, more familiar terminology: inputs are to outputs, as pre-images are to images.
              $endgroup$
              – Eevee Trainer
              2 hours ago














            0












            0








            0





            $begingroup$

            Think about what the inverse mapping for $f$ would be if indeed it would be a surjection: it would take the derivatives to the original function. Phrased another way, it would take a function to its antiderivative, right?



            Well, we know that all polynomials have an antiderivative (which is itself a polynomial), and since the codomain of $f$ is basically the set of real polynomials, we have a pre-image for each polynomial in $P$ - its antiderivative. Thus, $f$ is surjective.






            share|cite|improve this answer









            $endgroup$



            Think about what the inverse mapping for $f$ would be if indeed it would be a surjection: it would take the derivatives to the original function. Phrased another way, it would take a function to its antiderivative, right?



            Well, we know that all polynomials have an antiderivative (which is itself a polynomial), and since the codomain of $f$ is basically the set of real polynomials, we have a pre-image for each polynomial in $P$ - its antiderivative. Thus, $f$ is surjective.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 2 hours ago









            Eevee TrainerEevee Trainer

            10.5k31842




            10.5k31842












            • $begingroup$
              What do you mean by pre-image?
              $endgroup$
              – John Arg
              2 hours ago










            • $begingroup$
              You know how, say, $f(x)$ might be called the "image" for $x$? Same sort of deal: we would also say $x$ is the pre-image for $f(x)$. To use, arguably, more familiar terminology: inputs are to outputs, as pre-images are to images.
              $endgroup$
              – Eevee Trainer
              2 hours ago


















            • $begingroup$
              What do you mean by pre-image?
              $endgroup$
              – John Arg
              2 hours ago










            • $begingroup$
              You know how, say, $f(x)$ might be called the "image" for $x$? Same sort of deal: we would also say $x$ is the pre-image for $f(x)$. To use, arguably, more familiar terminology: inputs are to outputs, as pre-images are to images.
              $endgroup$
              – Eevee Trainer
              2 hours ago
















            $begingroup$
            What do you mean by pre-image?
            $endgroup$
            – John Arg
            2 hours ago




            $begingroup$
            What do you mean by pre-image?
            $endgroup$
            – John Arg
            2 hours ago












            $begingroup$
            You know how, say, $f(x)$ might be called the "image" for $x$? Same sort of deal: we would also say $x$ is the pre-image for $f(x)$. To use, arguably, more familiar terminology: inputs are to outputs, as pre-images are to images.
            $endgroup$
            – Eevee Trainer
            2 hours ago




            $begingroup$
            You know how, say, $f(x)$ might be called the "image" for $x$? Same sort of deal: we would also say $x$ is the pre-image for $f(x)$. To use, arguably, more familiar terminology: inputs are to outputs, as pre-images are to images.
            $endgroup$
            – Eevee Trainer
            2 hours ago


















            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3187852%2fdetermine-whether-f-is-a-function-an-injection-a-surjection%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

            How to label and detect the document text images

            Vallis Paradisi

            Tabula Rosettana