Expressing the converse, contra-positive, and inverse of conditional statements












0












$begingroup$


This problem is from Discrete Mathematics and its Applications



enter image description here



Here is my book's definition on converse, contrapositive, and inverse
enter image description here



And the common ways to express an implication



enter image description here



For this problem, is it better to translate the conditional statements that are not in the "if p, q form" to that form? That way everything will be consistent.



Going off the translation approach



27a. This conditional statement is already in the "if p, q form", so I don't have to do any translation




  • Converse - q -> p. If I ski tomorrow, it will snow today.

  • Contrapositive - ~q -> ~p. If I don't ski tomorrow, it will not snow today

  • Inverse - ~p -> ~q. If it doesn't snow today, I will not ski tomorrow.


27b. This conditional statement is in the "q whenever p" form so I translated that to "If there is going to be a quiz, I will come to class.




  • Converse - q -> p. If I come to class, there is going to be a quiz.

  • Contrapositive - ~q -> ~p. If I don't come to class there is not going to be a quiz.

  • Inverse - ~p -> ~q. If there is not going to be a quiz, I will not come to class.


27c. This conditional statement is in the p only if form, so I translated it to "if a positive integer is a prime, it has no divisors other than 1 and itself.




  • Converse - q -> p. If a positive integer has no divisors other than 1 and itself, it is prime.

  • Contrapositive - ~q -> ~p. If a positive integer has divisors other than 1 and itself, it is not prime.

  • Inverse - ~p -> ~q. If a positive integer is not prime, it has divisors other than 1 and itself.


Does look right/ sound logically coherent. Is it a good strategy to convert form to if p, q or is that not necessary?






discrete-mathematics logic propositional-calculus solution-verification







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    This problem is from Discrete Mathematics and its Applications



    enter image description here



    Here is my book's definition on converse, contrapositive, and inverse
    enter image description here



    And the common ways to express an implication



    enter image description here



    For this problem, is it better to translate the conditional statements that are not in the "if p, q form" to that form? That way everything will be consistent.



    Going off the translation approach



    27a. This conditional statement is already in the "if p, q form", so I don't have to do any translation




    • Converse - q -> p. If I ski tomorrow, it will snow today.

    • Contrapositive - ~q -> ~p. If I don't ski tomorrow, it will not snow today

    • Inverse - ~p -> ~q. If it doesn't snow today, I will not ski tomorrow.


    27b. This conditional statement is in the "q whenever p" form so I translated that to "If there is going to be a quiz, I will come to class.




    • Converse - q -> p. If I come to class, there is going to be a quiz.

    • Contrapositive - ~q -> ~p. If I don't come to class there is not going to be a quiz.

    • Inverse - ~p -> ~q. If there is not going to be a quiz, I will not come to class.


    27c. This conditional statement is in the p only if form, so I translated it to "if a positive integer is a prime, it has no divisors other than 1 and itself.




    • Converse - q -> p. If a positive integer has no divisors other than 1 and itself, it is prime.

    • Contrapositive - ~q -> ~p. If a positive integer has divisors other than 1 and itself, it is not prime.

    • Inverse - ~p -> ~q. If a positive integer is not prime, it has divisors other than 1 and itself.


    Does look right/ sound logically coherent. Is it a good strategy to convert form to if p, q or is that not necessary?






    discrete-mathematics logic propositional-calculus solution-verification







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      This problem is from Discrete Mathematics and its Applications



      enter image description here



      Here is my book's definition on converse, contrapositive, and inverse
      enter image description here



      And the common ways to express an implication



      enter image description here



      For this problem, is it better to translate the conditional statements that are not in the "if p, q form" to that form? That way everything will be consistent.



      Going off the translation approach



      27a. This conditional statement is already in the "if p, q form", so I don't have to do any translation




      • Converse - q -> p. If I ski tomorrow, it will snow today.

      • Contrapositive - ~q -> ~p. If I don't ski tomorrow, it will not snow today

      • Inverse - ~p -> ~q. If it doesn't snow today, I will not ski tomorrow.


      27b. This conditional statement is in the "q whenever p" form so I translated that to "If there is going to be a quiz, I will come to class.




      • Converse - q -> p. If I come to class, there is going to be a quiz.

      • Contrapositive - ~q -> ~p. If I don't come to class there is not going to be a quiz.

      • Inverse - ~p -> ~q. If there is not going to be a quiz, I will not come to class.


      27c. This conditional statement is in the p only if form, so I translated it to "if a positive integer is a prime, it has no divisors other than 1 and itself.




      • Converse - q -> p. If a positive integer has no divisors other than 1 and itself, it is prime.

      • Contrapositive - ~q -> ~p. If a positive integer has divisors other than 1 and itself, it is not prime.

      • Inverse - ~p -> ~q. If a positive integer is not prime, it has divisors other than 1 and itself.


      Does look right/ sound logically coherent. Is it a good strategy to convert form to if p, q or is that not necessary?






      discrete-mathematics logic propositional-calculus solution-verification







      share|cite|improve this question











      $endgroup$




      This problem is from Discrete Mathematics and its Applications



      enter image description here



      Here is my book's definition on converse, contrapositive, and inverse
      enter image description here



      And the common ways to express an implication



      enter image description here



      For this problem, is it better to translate the conditional statements that are not in the "if p, q form" to that form? That way everything will be consistent.



      Going off the translation approach



      27a. This conditional statement is already in the "if p, q form", so I don't have to do any translation




      • Converse - q -> p. If I ski tomorrow, it will snow today.

      • Contrapositive - ~q -> ~p. If I don't ski tomorrow, it will not snow today

      • Inverse - ~p -> ~q. If it doesn't snow today, I will not ski tomorrow.


      27b. This conditional statement is in the "q whenever p" form so I translated that to "If there is going to be a quiz, I will come to class.




      • Converse - q -> p. If I come to class, there is going to be a quiz.

      • Contrapositive - ~q -> ~p. If I don't come to class there is not going to be a quiz.

      • Inverse - ~p -> ~q. If there is not going to be a quiz, I will not come to class.


      27c. This conditional statement is in the p only if form, so I translated it to "if a positive integer is a prime, it has no divisors other than 1 and itself.




      • Converse - q -> p. If a positive integer has no divisors other than 1 and itself, it is prime.

      • Contrapositive - ~q -> ~p. If a positive integer has divisors other than 1 and itself, it is not prime.

      • Inverse - ~p -> ~q. If a positive integer is not prime, it has divisors other than 1 and itself.


      Does look right/ sound logically coherent. Is it a good strategy to convert form to if p, q or is that not necessary?






      discrete-mathematics logic propositional-calculus solution-verification




      discrete-mathematics logic propositional-calculus solution-verification






      share|cite|improve this question















      share|cite|improve this question













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      edited Jan 26 '15 at 0:06







      user147263

















      asked Jan 25 '15 at 22:08









      committedandroidercommittedandroider

      97341737




      97341737






















          1 Answer
          1






          active

          oldest

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          0












          $begingroup$


          1. You aren't entirely correct in your translations. In your example of 27, you keep "will" in the same part of the sentence, but they should be part of the proposition that you're moving around. I know, this seems small and pedantic, but to be completely concise and accurate, it's an important detail. I will mention, however, that your translation is probably acceptable for a discrete mathematics course, but in a pure logic course it wouldn't be appropriate.


          $p$ = "It snows today"



          $q$ = "I will ski tomorrow"



          So 27a converse should look like "If I will skill tomorrow, then it snows today."



          28 and 29 look to be just fine.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So I skill tomorrow isn't the same as I will ski tomorrow? To me they express the same thing an action(skiing) that will happen tomorrow.
            $endgroup$
            – committedandroider
            Jan 28 '15 at 20:32










          • $begingroup$
            @committedandroider I shall interpret "I ski" as having present tense to it. Thus, saying "I ski tomorrow" could perhaps serve as a shorthand for "I ski today and I will ski tomorrow." That differs from saying "I will ski tomorrow." (this isn't meant to imply anything about how people use a phrase like "I ski tomorrow".)
            $endgroup$
            – Doug Spoonwood
            Sep 25 '16 at 14:51










          • $begingroup$
            "If I will ski tomorrow, then it snows today" is not grammatical.
            $endgroup$
            – Fabio Somenzi
            Jan 6 at 4:19











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          1 Answer
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          active

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          active

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          0












          $begingroup$


          1. You aren't entirely correct in your translations. In your example of 27, you keep "will" in the same part of the sentence, but they should be part of the proposition that you're moving around. I know, this seems small and pedantic, but to be completely concise and accurate, it's an important detail. I will mention, however, that your translation is probably acceptable for a discrete mathematics course, but in a pure logic course it wouldn't be appropriate.


          $p$ = "It snows today"



          $q$ = "I will ski tomorrow"



          So 27a converse should look like "If I will skill tomorrow, then it snows today."



          28 and 29 look to be just fine.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So I skill tomorrow isn't the same as I will ski tomorrow? To me they express the same thing an action(skiing) that will happen tomorrow.
            $endgroup$
            – committedandroider
            Jan 28 '15 at 20:32










          • $begingroup$
            @committedandroider I shall interpret "I ski" as having present tense to it. Thus, saying "I ski tomorrow" could perhaps serve as a shorthand for "I ski today and I will ski tomorrow." That differs from saying "I will ski tomorrow." (this isn't meant to imply anything about how people use a phrase like "I ski tomorrow".)
            $endgroup$
            – Doug Spoonwood
            Sep 25 '16 at 14:51










          • $begingroup$
            "If I will ski tomorrow, then it snows today" is not grammatical.
            $endgroup$
            – Fabio Somenzi
            Jan 6 at 4:19
















          0












          $begingroup$


          1. You aren't entirely correct in your translations. In your example of 27, you keep "will" in the same part of the sentence, but they should be part of the proposition that you're moving around. I know, this seems small and pedantic, but to be completely concise and accurate, it's an important detail. I will mention, however, that your translation is probably acceptable for a discrete mathematics course, but in a pure logic course it wouldn't be appropriate.


          $p$ = "It snows today"



          $q$ = "I will ski tomorrow"



          So 27a converse should look like "If I will skill tomorrow, then it snows today."



          28 and 29 look to be just fine.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So I skill tomorrow isn't the same as I will ski tomorrow? To me they express the same thing an action(skiing) that will happen tomorrow.
            $endgroup$
            – committedandroider
            Jan 28 '15 at 20:32










          • $begingroup$
            @committedandroider I shall interpret "I ski" as having present tense to it. Thus, saying "I ski tomorrow" could perhaps serve as a shorthand for "I ski today and I will ski tomorrow." That differs from saying "I will ski tomorrow." (this isn't meant to imply anything about how people use a phrase like "I ski tomorrow".)
            $endgroup$
            – Doug Spoonwood
            Sep 25 '16 at 14:51










          • $begingroup$
            "If I will ski tomorrow, then it snows today" is not grammatical.
            $endgroup$
            – Fabio Somenzi
            Jan 6 at 4:19














          0












          0








          0





          $begingroup$


          1. You aren't entirely correct in your translations. In your example of 27, you keep "will" in the same part of the sentence, but they should be part of the proposition that you're moving around. I know, this seems small and pedantic, but to be completely concise and accurate, it's an important detail. I will mention, however, that your translation is probably acceptable for a discrete mathematics course, but in a pure logic course it wouldn't be appropriate.


          $p$ = "It snows today"



          $q$ = "I will ski tomorrow"



          So 27a converse should look like "If I will skill tomorrow, then it snows today."



          28 and 29 look to be just fine.






          share|cite|improve this answer









          $endgroup$




          1. You aren't entirely correct in your translations. In your example of 27, you keep "will" in the same part of the sentence, but they should be part of the proposition that you're moving around. I know, this seems small and pedantic, but to be completely concise and accurate, it's an important detail. I will mention, however, that your translation is probably acceptable for a discrete mathematics course, but in a pure logic course it wouldn't be appropriate.


          $p$ = "It snows today"



          $q$ = "I will ski tomorrow"



          So 27a converse should look like "If I will skill tomorrow, then it snows today."



          28 and 29 look to be just fine.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 25 '15 at 22:49









          ProbablyNotYourProfessorProbablyNotYourProfessor

          516




          516












          • $begingroup$
            So I skill tomorrow isn't the same as I will ski tomorrow? To me they express the same thing an action(skiing) that will happen tomorrow.
            $endgroup$
            – committedandroider
            Jan 28 '15 at 20:32










          • $begingroup$
            @committedandroider I shall interpret "I ski" as having present tense to it. Thus, saying "I ski tomorrow" could perhaps serve as a shorthand for "I ski today and I will ski tomorrow." That differs from saying "I will ski tomorrow." (this isn't meant to imply anything about how people use a phrase like "I ski tomorrow".)
            $endgroup$
            – Doug Spoonwood
            Sep 25 '16 at 14:51










          • $begingroup$
            "If I will ski tomorrow, then it snows today" is not grammatical.
            $endgroup$
            – Fabio Somenzi
            Jan 6 at 4:19


















          • $begingroup$
            So I skill tomorrow isn't the same as I will ski tomorrow? To me they express the same thing an action(skiing) that will happen tomorrow.
            $endgroup$
            – committedandroider
            Jan 28 '15 at 20:32










          • $begingroup$
            @committedandroider I shall interpret "I ski" as having present tense to it. Thus, saying "I ski tomorrow" could perhaps serve as a shorthand for "I ski today and I will ski tomorrow." That differs from saying "I will ski tomorrow." (this isn't meant to imply anything about how people use a phrase like "I ski tomorrow".)
            $endgroup$
            – Doug Spoonwood
            Sep 25 '16 at 14:51










          • $begingroup$
            "If I will ski tomorrow, then it snows today" is not grammatical.
            $endgroup$
            – Fabio Somenzi
            Jan 6 at 4:19
















          $begingroup$
          So I skill tomorrow isn't the same as I will ski tomorrow? To me they express the same thing an action(skiing) that will happen tomorrow.
          $endgroup$
          – committedandroider
          Jan 28 '15 at 20:32




          $begingroup$
          So I skill tomorrow isn't the same as I will ski tomorrow? To me they express the same thing an action(skiing) that will happen tomorrow.
          $endgroup$
          – committedandroider
          Jan 28 '15 at 20:32












          $begingroup$
          @committedandroider I shall interpret "I ski" as having present tense to it. Thus, saying "I ski tomorrow" could perhaps serve as a shorthand for "I ski today and I will ski tomorrow." That differs from saying "I will ski tomorrow." (this isn't meant to imply anything about how people use a phrase like "I ski tomorrow".)
          $endgroup$
          – Doug Spoonwood
          Sep 25 '16 at 14:51




          $begingroup$
          @committedandroider I shall interpret "I ski" as having present tense to it. Thus, saying "I ski tomorrow" could perhaps serve as a shorthand for "I ski today and I will ski tomorrow." That differs from saying "I will ski tomorrow." (this isn't meant to imply anything about how people use a phrase like "I ski tomorrow".)
          $endgroup$
          – Doug Spoonwood
          Sep 25 '16 at 14:51












          $begingroup$
          "If I will ski tomorrow, then it snows today" is not grammatical.
          $endgroup$
          – Fabio Somenzi
          Jan 6 at 4:19




          $begingroup$
          "If I will ski tomorrow, then it snows today" is not grammatical.
          $endgroup$
          – Fabio Somenzi
          Jan 6 at 4:19


















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