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lgbasallote Group Title

Logic Question: A positive integer is prime only if it has no other divisors other than 1 and itself. Why is the p here "A positive integer is prime" and not "it has no other divisors other than 1 and itself."

  • 2 years ago
  • 2 years ago

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  1. joemath314159 Group Title
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    So we dont have to worry about negative numbers.

    • 2 years ago
  2. lgbasallote Group Title
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    how does that relate?

    • 2 years ago
  3. lgbasallote Group Title
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    by p im referring to p and q by the way something like \(p \rightarrow q\)

    • 2 years ago
  4. wio Group Title
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    What p?

    • 2 years ago
  5. farmdawgnation Group Title
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    The p is the conditional.

    • 2 years ago
  6. lgbasallote Group Title
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    p succeeds the "if" right?

    • 2 years ago
  7. farmdawgnation Group Title
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    So you'd rewrite this as, If positive integer n is prime, it has no other divisors other than 1 and itself. However, you CAN'T say.. If n has no other divisors than 1 and itself, it is a positive integer.

    • 2 years ago
  8. farmdawgnation Group Title
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    That'd be a problem.

    • 2 years ago
  9. wio Group Title
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    "only if" works like "if" in reverse.

    • 2 years ago
  10. joemath314159 Group Title
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    ah, i misunderstood. I would read that statement as: If a number has no divisors other than one and itself, then it is prime.

    • 2 years ago
  11. lgbasallote Group Title
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    ...i didn't know that....

    • 2 years ago
  12. lgbasallote Group Title
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    are there any other phrases that work as if in reverse?

    • 2 years ago
  13. wio Group Title
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    A if B becomes B only if A

    • 2 years ago
  14. wio Group Title
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    A is necessary for B means \(B\rightarrow A\). A is sufficient for B means \(A\rightarrow B\). A if B means \(B\rightarrow A\) A only if B means \(A\rightarrow B\)

    • 2 years ago
  15. lgbasallote Group Title
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    ahh just what i needed. what about necessary but not sufficient?

    • 2 years ago
  16. wio Group Title
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    necessary but not sufficient basically is the same as necessary.

    • 2 years ago
  17. wio Group Title
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    necessary and sufficient is bi-conditional.

    • 2 years ago
  18. wio Group Title
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    The best thing to do is use 'rain' and 'clouds' and see if it makes sense. Then remember that rain implies clouds.

    • 2 years ago
  19. lgbasallote Group Title
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    \[(B \rightarrow A) \wedge (\neg A \rightarrow B)\] like that?

    • 2 years ago
  20. wio Group Title
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    but clouds don't imply rain.

    • 2 years ago
  21. wio Group Title
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    No, not like that.

    • 2 years ago
  22. lgbasallote Group Title
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    because i remember something that said necessary but not sufficient and had this solution \[(q \rightarrow (\neg r \wedge \neg p)) \wedge \neg((\neg r \wedge \neg p) \rightarrow q)\]

    • 2 years ago
  23. lgbasallote Group Title
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    the statement was this: For hiking to be safe, it is necessary but not sufficient that berries be not ripe along the trail and for grizzly bears not to have been seen in the area

    • 2 years ago
  24. wio Group Title
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    I mean you could perhaps say \((B\rightarrow A) \wedge \neg (A\rightarrow B)\)

    • 2 years ago
  25. lgbasallote Group Title
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    the p, q, r are these: p: Grizzly bears have been sean in the area q: hiking is safe r: berries are ripe along the trail

    • 2 years ago
  26. lgbasallote Group Title
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    so any explanations for that?

    • 2 years ago
  27. wio Group Title
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    Well if \(B\rightarrow A\) means necessary and \(A\rightarrow B\) means sufficient then necessary but not sufficient could be written as \(B\rightarrow A)\wedge \neg (A\rightarrow B\))

    • 2 years ago
  28. lgbasallote Group Title
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    ahh yes. makes sense

    • 2 years ago
  29. wio Group Title
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    But there is a big of an ambiguity for me, because when they say necessary and not sufficient, do they mean necessary but not necessarily sufficient, or necessary and never sufficient.

    • 2 years ago
  30. lgbasallote Group Title
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    that's why i hate math.....

    • 2 years ago
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