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princesspixie

Write the contrapositive of the conditional statement below. “ n is a prime number implies that n=2 or n is an odd number.” Your answer must contain a conjunction in its premise.

  • one year ago
  • one year ago

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  1. KingGeorge
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    So in general, the contrapositive of the statement (\(\,p\implies q\)) is \((\neg \,q\implies \neg \,p\)). In this case, both your p and q are other sentences. Can you tell me what the "p" is in your conditional statement?

    • one year ago
  2. princesspixie
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    n is a prime number

    • one year ago
  3. KingGeorge
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    Right. And the "q"?

    • one year ago
  4. princesspixie
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    n is an odd number

    • one year ago
  5. KingGeorge
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    Almost. The "q" in your statement is "n=2 or n is an odd number." Notice that there's an "or" in this statement, so even this can be broken down into something like "q OR r," where q is "n=2", and r is "n is an odd number."

    • one year ago
  6. princesspixie
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    oh i

    • one year ago
  7. princesspixie
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    SEE

    • one year ago
  8. KingGeorge
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    So the statement "n is a prime number implies that n=2 or n is an odd number" can be rewritten as \[p\implies (q \vee r)\]

    • one year ago
  9. princesspixie
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    if n is not a odd number then n is not a prime number or n = 2

    • one year ago
  10. KingGeorge
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    You're very close, and certainly have the right idea. The contrapositive of \(p\implies (q\vee r)\) is \(\neg(q\vee r)\implies \neg p\). Can you tell me what \(\neg(q\vee r)\) can be rewritten as?

    • one year ago
  11. princesspixie
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    not sure

    • one year ago
  12. KingGeorge
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    This is a thing called "De Morgan's Law." It says that \[\neg(q\vee r) \Longleftrightarrow (\neg \,q) \wedge(\neg\,r) \]Can you translate this back into our statements?

    • one year ago
  13. princesspixie
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    im confused

    • one year ago
  14. KingGeorge
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    So if q is "n=2," what is ~q?

    • one year ago
  15. princesspixie
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    not =2

    • one year ago
  16. princesspixie
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    |dw:1355893931180:dw|

    • one year ago
  17. KingGeorge
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    Right. And if r is "n is an odd number," what is ~r?

    • one year ago
  18. princesspixie
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    n is not an odd number

    • one year ago
  19. KingGeorge
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    Bingo. So if we then combine those two statements, ~q AND ~r can be written as "n is not 2, and n is not an odd number." Make sense?

    • one year ago
  20. princesspixie
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    yes

    • one year ago
  21. KingGeorge
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    Great. Then the contrapositive of your original statement should be "If n is not 2, and n is not an odd number, then n is not prime."

    • one year ago
  22. princesspixie
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    that the answer it was that simple?

    • one year ago
  23. KingGeorge
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    That was it. Did it all make sense?

    • one year ago
  24. KingGeorge
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    You almost had it the first time, you just got a little mixed up with the second half of the statement.

    • one year ago
  25. princesspixie
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    yes but what does it mean by Your answer must contain a conjunction in its premise.

    • one year ago
  26. KingGeorge
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    That just means that it has to have "and" somewhere in the statement. So the way it's written, it's fine. If we had written it as "If n is an even integer greater than two, then n is not prime" instead, which says the same thing, it would not be the correct solution because it does not contain "and."

    • one year ago
  27. princesspixie
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    oh i see thanks so much!!

    • one year ago
  28. KingGeorge
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    You're welcome.

    • one year ago
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