Why is this a proof by contradiction for this algorithm? Isn't this a direct proof instead?












3












$begingroup$


First Slide: Find Max(A)




  1. // INPUT: A[1..n] - an array of integers

  2. // OUTPUT: an element m of A such that m >= A[j], for all 1 <= j <= A.length

  3. max = A[j==1]

  4. for j = 2 to A.length

  5. if max < A[j]

  6. max = A[j]

  7. return max


Second Slide: Proof By Contradiction



Proof: Suppose the algorithm is incorrect. Then for some input A, either:




  1. max is not an element of A or

  2. A has an element A[j] such that max < A[j]


Max is initialized to and assigned to elements of A - so (1) is impossible.



After the j-th iteration of the for loop (lines 4 - 6), max >= A[j]. From lines 5,6 max only increases. Therefore upon termination, max >= A[j] which contradicts (2).



End Of Slides



This algorithm (first slide) finds the max element in the array. This is a proof by contradiction. Isn't this algorithm already proved when it gets to max >= A[j] in the last line of the second slide. Is which contradicts (2) even necessary? Because in the second slide you showed max is in the array, and then you met the condition of a maxium m: an element m of A such
that m >= A[j] for all 1 <= j <= A.length.



This seems to be two proofs in one. And since it seems to me that the direct proof happens first, then the proof by contradiction is redundant and therefore not necessary. Am i missing something here? Is this only a proof by contradiction? Or is it a direct proof instead?



The image below is just the slides 1 and 2 that I transcribed above. From York University.



Find Max: Slides 1 and 2










share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    The proof assumes the opposite of what is to be proved then shows a contradiction. So it is a proof by contradiction. And so it is not a direct proof. It happens to prove that the algorithm is correct. "when it gets to max >= A[j]" it has gotten there by having assumed the opposite of what is to be proved; it has not proved "max >= A[j]" having assumed nothing. Indeed since it assumed something that happens to be false, anything can be derived thereafter before discharging that assumption.
    $endgroup$
    – philipxy
    Mar 25 at 8:46








  • 1




    $begingroup$
    Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics (note that you can use LaTeX) and don't forget to give proper attribution to your sources!
    $endgroup$
    – dkaeae
    Mar 25 at 10:22
















3












$begingroup$


First Slide: Find Max(A)




  1. // INPUT: A[1..n] - an array of integers

  2. // OUTPUT: an element m of A such that m >= A[j], for all 1 <= j <= A.length

  3. max = A[j==1]

  4. for j = 2 to A.length

  5. if max < A[j]

  6. max = A[j]

  7. return max


Second Slide: Proof By Contradiction



Proof: Suppose the algorithm is incorrect. Then for some input A, either:




  1. max is not an element of A or

  2. A has an element A[j] such that max < A[j]


Max is initialized to and assigned to elements of A - so (1) is impossible.



After the j-th iteration of the for loop (lines 4 - 6), max >= A[j]. From lines 5,6 max only increases. Therefore upon termination, max >= A[j] which contradicts (2).



End Of Slides



This algorithm (first slide) finds the max element in the array. This is a proof by contradiction. Isn't this algorithm already proved when it gets to max >= A[j] in the last line of the second slide. Is which contradicts (2) even necessary? Because in the second slide you showed max is in the array, and then you met the condition of a maxium m: an element m of A such
that m >= A[j] for all 1 <= j <= A.length.



This seems to be two proofs in one. And since it seems to me that the direct proof happens first, then the proof by contradiction is redundant and therefore not necessary. Am i missing something here? Is this only a proof by contradiction? Or is it a direct proof instead?



The image below is just the slides 1 and 2 that I transcribed above. From York University.



Find Max: Slides 1 and 2










share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    The proof assumes the opposite of what is to be proved then shows a contradiction. So it is a proof by contradiction. And so it is not a direct proof. It happens to prove that the algorithm is correct. "when it gets to max >= A[j]" it has gotten there by having assumed the opposite of what is to be proved; it has not proved "max >= A[j]" having assumed nothing. Indeed since it assumed something that happens to be false, anything can be derived thereafter before discharging that assumption.
    $endgroup$
    – philipxy
    Mar 25 at 8:46








  • 1




    $begingroup$
    Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics (note that you can use LaTeX) and don't forget to give proper attribution to your sources!
    $endgroup$
    – dkaeae
    Mar 25 at 10:22














3












3








3





$begingroup$


First Slide: Find Max(A)




  1. // INPUT: A[1..n] - an array of integers

  2. // OUTPUT: an element m of A such that m >= A[j], for all 1 <= j <= A.length

  3. max = A[j==1]

  4. for j = 2 to A.length

  5. if max < A[j]

  6. max = A[j]

  7. return max


Second Slide: Proof By Contradiction



Proof: Suppose the algorithm is incorrect. Then for some input A, either:




  1. max is not an element of A or

  2. A has an element A[j] such that max < A[j]


Max is initialized to and assigned to elements of A - so (1) is impossible.



After the j-th iteration of the for loop (lines 4 - 6), max >= A[j]. From lines 5,6 max only increases. Therefore upon termination, max >= A[j] which contradicts (2).



End Of Slides



This algorithm (first slide) finds the max element in the array. This is a proof by contradiction. Isn't this algorithm already proved when it gets to max >= A[j] in the last line of the second slide. Is which contradicts (2) even necessary? Because in the second slide you showed max is in the array, and then you met the condition of a maxium m: an element m of A such
that m >= A[j] for all 1 <= j <= A.length.



This seems to be two proofs in one. And since it seems to me that the direct proof happens first, then the proof by contradiction is redundant and therefore not necessary. Am i missing something here? Is this only a proof by contradiction? Or is it a direct proof instead?



The image below is just the slides 1 and 2 that I transcribed above. From York University.



Find Max: Slides 1 and 2










share|cite|improve this question











$endgroup$




First Slide: Find Max(A)




  1. // INPUT: A[1..n] - an array of integers

  2. // OUTPUT: an element m of A such that m >= A[j], for all 1 <= j <= A.length

  3. max = A[j==1]

  4. for j = 2 to A.length

  5. if max < A[j]

  6. max = A[j]

  7. return max


Second Slide: Proof By Contradiction



Proof: Suppose the algorithm is incorrect. Then for some input A, either:




  1. max is not an element of A or

  2. A has an element A[j] such that max < A[j]


Max is initialized to and assigned to elements of A - so (1) is impossible.



After the j-th iteration of the for loop (lines 4 - 6), max >= A[j]. From lines 5,6 max only increases. Therefore upon termination, max >= A[j] which contradicts (2).



End Of Slides



This algorithm (first slide) finds the max element in the array. This is a proof by contradiction. Isn't this algorithm already proved when it gets to max >= A[j] in the last line of the second slide. Is which contradicts (2) even necessary? Because in the second slide you showed max is in the array, and then you met the condition of a maxium m: an element m of A such
that m >= A[j] for all 1 <= j <= A.length.



This seems to be two proofs in one. And since it seems to me that the direct proof happens first, then the proof by contradiction is redundant and therefore not necessary. Am i missing something here? Is this only a proof by contradiction? Or is it a direct proof instead?



The image below is just the slides 1 and 2 that I transcribed above. From York University.



Find Max: Slides 1 and 2







algorithms correctness-proof check-my-answer






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 26 at 1:25







user100752

















asked Mar 25 at 2:40









user100752user100752

1814




1814








  • 3




    $begingroup$
    The proof assumes the opposite of what is to be proved then shows a contradiction. So it is a proof by contradiction. And so it is not a direct proof. It happens to prove that the algorithm is correct. "when it gets to max >= A[j]" it has gotten there by having assumed the opposite of what is to be proved; it has not proved "max >= A[j]" having assumed nothing. Indeed since it assumed something that happens to be false, anything can be derived thereafter before discharging that assumption.
    $endgroup$
    – philipxy
    Mar 25 at 8:46








  • 1




    $begingroup$
    Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics (note that you can use LaTeX) and don't forget to give proper attribution to your sources!
    $endgroup$
    – dkaeae
    Mar 25 at 10:22














  • 3




    $begingroup$
    The proof assumes the opposite of what is to be proved then shows a contradiction. So it is a proof by contradiction. And so it is not a direct proof. It happens to prove that the algorithm is correct. "when it gets to max >= A[j]" it has gotten there by having assumed the opposite of what is to be proved; it has not proved "max >= A[j]" having assumed nothing. Indeed since it assumed something that happens to be false, anything can be derived thereafter before discharging that assumption.
    $endgroup$
    – philipxy
    Mar 25 at 8:46








  • 1




    $begingroup$
    Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics (note that you can use LaTeX) and don't forget to give proper attribution to your sources!
    $endgroup$
    – dkaeae
    Mar 25 at 10:22








3




3




$begingroup$
The proof assumes the opposite of what is to be proved then shows a contradiction. So it is a proof by contradiction. And so it is not a direct proof. It happens to prove that the algorithm is correct. "when it gets to max >= A[j]" it has gotten there by having assumed the opposite of what is to be proved; it has not proved "max >= A[j]" having assumed nothing. Indeed since it assumed something that happens to be false, anything can be derived thereafter before discharging that assumption.
$endgroup$
– philipxy
Mar 25 at 8:46






$begingroup$
The proof assumes the opposite of what is to be proved then shows a contradiction. So it is a proof by contradiction. And so it is not a direct proof. It happens to prove that the algorithm is correct. "when it gets to max >= A[j]" it has gotten there by having assumed the opposite of what is to be proved; it has not proved "max >= A[j]" having assumed nothing. Indeed since it assumed something that happens to be false, anything can be derived thereafter before discharging that assumption.
$endgroup$
– philipxy
Mar 25 at 8:46






1




1




$begingroup$
Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics (note that you can use LaTeX) and don't forget to give proper attribution to your sources!
$endgroup$
– dkaeae
Mar 25 at 10:22




$begingroup$
Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics (note that you can use LaTeX) and don't forget to give proper attribution to your sources!
$endgroup$
– dkaeae
Mar 25 at 10:22










2 Answers
2






active

oldest

votes


















4












$begingroup$

You seem to think the structure of the proof is:




  1. suppose the algorithm is incorrect;

  2. prove that the algorithm is, in fact, correct;

  3. this contradicts 1., so the algorithm is correct.


That's almost, but not quite true. Step 2 doesn't prove that the returned value $mathrm{max}$ is bigger than every element of the array, which is what would be required to prove that the algorithm is correct. It just proves that $mathrm{max}$ is bigger than the specific array value that was supposed to be a counterexample in step 1.



But, as Yuval has pointed out, the proof is much easier if you forget about contradiction completely and prove that $mathrm{max}$ is in the array (identical to the given proof) and then show that $mathrm{max}$ is at least as big as every array element (identical to the proof except "for all $j$" instead of just for one supposed counterexample.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    When it says "after the j-th iteration of the for-loop (lines 4 - 6) max >= A[j]" is it talking specifically about the index of the assumed counter example A[j]? When I initially read this proof I was thinking about j taking on all the values, so I was thinking at the end of each iteration max >= A[j]. And this is why I guess I was thinking along the lines of a direct proof.
    $endgroup$
    – user100752
    Mar 27 at 6:03












  • $begingroup$
    @user100752 Yes, it's specifically the $j$ in point 2.
    $endgroup$
    – David Richerby
    Mar 27 at 10:10



















3












$begingroup$

It's a proof by contradiction that could easily be rewritten as a direct proof.



To rephrase it as a direct proof, we divide it into two claims:





  1. $max$ is an element of $A$.


  2. $max geq A[j]$ for all $j$.


We can conclude that $max = max(A)$.






share|cite|improve this answer









$endgroup$














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    2 Answers
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    4












    $begingroup$

    You seem to think the structure of the proof is:




    1. suppose the algorithm is incorrect;

    2. prove that the algorithm is, in fact, correct;

    3. this contradicts 1., so the algorithm is correct.


    That's almost, but not quite true. Step 2 doesn't prove that the returned value $mathrm{max}$ is bigger than every element of the array, which is what would be required to prove that the algorithm is correct. It just proves that $mathrm{max}$ is bigger than the specific array value that was supposed to be a counterexample in step 1.



    But, as Yuval has pointed out, the proof is much easier if you forget about contradiction completely and prove that $mathrm{max}$ is in the array (identical to the given proof) and then show that $mathrm{max}$ is at least as big as every array element (identical to the proof except "for all $j$" instead of just for one supposed counterexample.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      When it says "after the j-th iteration of the for-loop (lines 4 - 6) max >= A[j]" is it talking specifically about the index of the assumed counter example A[j]? When I initially read this proof I was thinking about j taking on all the values, so I was thinking at the end of each iteration max >= A[j]. And this is why I guess I was thinking along the lines of a direct proof.
      $endgroup$
      – user100752
      Mar 27 at 6:03












    • $begingroup$
      @user100752 Yes, it's specifically the $j$ in point 2.
      $endgroup$
      – David Richerby
      Mar 27 at 10:10
















    4












    $begingroup$

    You seem to think the structure of the proof is:




    1. suppose the algorithm is incorrect;

    2. prove that the algorithm is, in fact, correct;

    3. this contradicts 1., so the algorithm is correct.


    That's almost, but not quite true. Step 2 doesn't prove that the returned value $mathrm{max}$ is bigger than every element of the array, which is what would be required to prove that the algorithm is correct. It just proves that $mathrm{max}$ is bigger than the specific array value that was supposed to be a counterexample in step 1.



    But, as Yuval has pointed out, the proof is much easier if you forget about contradiction completely and prove that $mathrm{max}$ is in the array (identical to the given proof) and then show that $mathrm{max}$ is at least as big as every array element (identical to the proof except "for all $j$" instead of just for one supposed counterexample.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      When it says "after the j-th iteration of the for-loop (lines 4 - 6) max >= A[j]" is it talking specifically about the index of the assumed counter example A[j]? When I initially read this proof I was thinking about j taking on all the values, so I was thinking at the end of each iteration max >= A[j]. And this is why I guess I was thinking along the lines of a direct proof.
      $endgroup$
      – user100752
      Mar 27 at 6:03












    • $begingroup$
      @user100752 Yes, it's specifically the $j$ in point 2.
      $endgroup$
      – David Richerby
      Mar 27 at 10:10














    4












    4








    4





    $begingroup$

    You seem to think the structure of the proof is:




    1. suppose the algorithm is incorrect;

    2. prove that the algorithm is, in fact, correct;

    3. this contradicts 1., so the algorithm is correct.


    That's almost, but not quite true. Step 2 doesn't prove that the returned value $mathrm{max}$ is bigger than every element of the array, which is what would be required to prove that the algorithm is correct. It just proves that $mathrm{max}$ is bigger than the specific array value that was supposed to be a counterexample in step 1.



    But, as Yuval has pointed out, the proof is much easier if you forget about contradiction completely and prove that $mathrm{max}$ is in the array (identical to the given proof) and then show that $mathrm{max}$ is at least as big as every array element (identical to the proof except "for all $j$" instead of just for one supposed counterexample.






    share|cite|improve this answer









    $endgroup$



    You seem to think the structure of the proof is:




    1. suppose the algorithm is incorrect;

    2. prove that the algorithm is, in fact, correct;

    3. this contradicts 1., so the algorithm is correct.


    That's almost, but not quite true. Step 2 doesn't prove that the returned value $mathrm{max}$ is bigger than every element of the array, which is what would be required to prove that the algorithm is correct. It just proves that $mathrm{max}$ is bigger than the specific array value that was supposed to be a counterexample in step 1.



    But, as Yuval has pointed out, the proof is much easier if you forget about contradiction completely and prove that $mathrm{max}$ is in the array (identical to the given proof) and then show that $mathrm{max}$ is at least as big as every array element (identical to the proof except "for all $j$" instead of just for one supposed counterexample.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Mar 25 at 8:51









    David RicherbyDavid Richerby

    69.6k15106195




    69.6k15106195












    • $begingroup$
      When it says "after the j-th iteration of the for-loop (lines 4 - 6) max >= A[j]" is it talking specifically about the index of the assumed counter example A[j]? When I initially read this proof I was thinking about j taking on all the values, so I was thinking at the end of each iteration max >= A[j]. And this is why I guess I was thinking along the lines of a direct proof.
      $endgroup$
      – user100752
      Mar 27 at 6:03












    • $begingroup$
      @user100752 Yes, it's specifically the $j$ in point 2.
      $endgroup$
      – David Richerby
      Mar 27 at 10:10


















    • $begingroup$
      When it says "after the j-th iteration of the for-loop (lines 4 - 6) max >= A[j]" is it talking specifically about the index of the assumed counter example A[j]? When I initially read this proof I was thinking about j taking on all the values, so I was thinking at the end of each iteration max >= A[j]. And this is why I guess I was thinking along the lines of a direct proof.
      $endgroup$
      – user100752
      Mar 27 at 6:03












    • $begingroup$
      @user100752 Yes, it's specifically the $j$ in point 2.
      $endgroup$
      – David Richerby
      Mar 27 at 10:10
















    $begingroup$
    When it says "after the j-th iteration of the for-loop (lines 4 - 6) max >= A[j]" is it talking specifically about the index of the assumed counter example A[j]? When I initially read this proof I was thinking about j taking on all the values, so I was thinking at the end of each iteration max >= A[j]. And this is why I guess I was thinking along the lines of a direct proof.
    $endgroup$
    – user100752
    Mar 27 at 6:03






    $begingroup$
    When it says "after the j-th iteration of the for-loop (lines 4 - 6) max >= A[j]" is it talking specifically about the index of the assumed counter example A[j]? When I initially read this proof I was thinking about j taking on all the values, so I was thinking at the end of each iteration max >= A[j]. And this is why I guess I was thinking along the lines of a direct proof.
    $endgroup$
    – user100752
    Mar 27 at 6:03














    $begingroup$
    @user100752 Yes, it's specifically the $j$ in point 2.
    $endgroup$
    – David Richerby
    Mar 27 at 10:10




    $begingroup$
    @user100752 Yes, it's specifically the $j$ in point 2.
    $endgroup$
    – David Richerby
    Mar 27 at 10:10











    3












    $begingroup$

    It's a proof by contradiction that could easily be rewritten as a direct proof.



    To rephrase it as a direct proof, we divide it into two claims:





    1. $max$ is an element of $A$.


    2. $max geq A[j]$ for all $j$.


    We can conclude that $max = max(A)$.






    share|cite|improve this answer









    $endgroup$


















      3












      $begingroup$

      It's a proof by contradiction that could easily be rewritten as a direct proof.



      To rephrase it as a direct proof, we divide it into two claims:





      1. $max$ is an element of $A$.


      2. $max geq A[j]$ for all $j$.


      We can conclude that $max = max(A)$.






      share|cite|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        It's a proof by contradiction that could easily be rewritten as a direct proof.



        To rephrase it as a direct proof, we divide it into two claims:





        1. $max$ is an element of $A$.


        2. $max geq A[j]$ for all $j$.


        We can conclude that $max = max(A)$.






        share|cite|improve this answer









        $endgroup$



        It's a proof by contradiction that could easily be rewritten as a direct proof.



        To rephrase it as a direct proof, we divide it into two claims:





        1. $max$ is an element of $A$.


        2. $max geq A[j]$ for all $j$.


        We can conclude that $max = max(A)$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 25 at 7:36









        Yuval FilmusYuval Filmus

        196k15184349




        196k15184349






























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