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princesspixie

  • 2 years ago

please tell me if this is correct: “All mathematicians must be good logicians and all good logicians must justify their claims.”

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  1. princesspixie
    • 2 years ago
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    a) Write this statement symbolically as a conjunction of two conditional statements. Use three components (p, q, and r) and explicitly state what these are in your work. Remember: a component in logic is a simple declarative sentence that is either true or false. For example, "You are a student in MAT 101" would be a component statement. Note how this sentence contains no logical connectives like "not", "or", "and", etc. b) Write the negation of this statement symbolically. Explain your reasoning here. c) Now use part b) to write the negation of this statement verbally. A- p means 'person is mathematician' q means 'person is good logician' and r means 'justifies their claims' (p⟹q)∧(q⟹r) B- (~p⟹~q)∧(~q⟹~r) You put the ~ mark because that negates the statement. C - If a person is not a mathematician then the person is not a good logician or they do not justify their claims

  2. princesspixie
    • 2 years ago
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    The Boxes are arrows

  3. satellite73
    • 2 years ago
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    p: a person is a good mathematician q: a person is a good logician \(p\implies q\) : if a person is a good mathematician, then a person is a good logician rather tortured english, would probably say if a person is a good mathematician then he or she is also a good logician

  4. princesspixie
    • 2 years ago
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    what would the r statement be?

  5. satellite73
    • 2 years ago
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    \(q\implies r\) : if a person is a good logician, then they justify their claims

  6. princesspixie
    • 2 years ago
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    would i say r: person justifies their claims

  7. satellite73
    • 2 years ago
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    \[(p\implies q)\land (q\implies r)\] take those two statements and stick the word "and" between them

  8. princesspixie
    • 2 years ago
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    what do you mean those two statements and stick an and?

  9. princesspixie
    • 2 years ago
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    ^ is the symbol for and?

  10. satellite73
    • 2 years ago
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    yes

  11. princesspixie
    • 2 years ago
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    ok is the negation correct?

  12. satellite73
    • 2 years ago
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    if a person is a good logician, then they justify their claims and if a person is a good logician, then they justify their claims

  13. princesspixie
    • 2 years ago
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    thats the negation ?

  14. satellite73
    • 2 years ago
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    no that is \[(p\implies q)\land (q\implies r)\]

  15. princesspixie
    • 2 years ago
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    ok would I see.. would I put a ~ before the q and the r to negate it ?

  16. princesspixie
    • 2 years ago
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    or infront of all the letters?

  17. satellite73
    • 2 years ago
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    no you have to put \(\lnot\) in front of the whole thing

  18. princesspixie
    • 2 years ago
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    ¬(p⟹q)∧(q⟹r) ?

  19. princesspixie
    • 2 years ago
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    what is the explanation for that ?

  20. satellite73
    • 2 years ago
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    \[\lnot\left((p\implies q)\land (q\implies r)\right)\] then you should rewrite without the \(\implies \) and use demorgan laws

  21. princesspixie
    • 2 years ago
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    I dont understand..

  22. satellite73
    • 2 years ago
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    it will take a while do you know that \(p\implies q\equiv \lnot p\lor q\)?

  23. princesspixie
    • 2 years ago
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    No..

  24. satellite73
    • 2 years ago
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    to negate a statement you put a big \(\lnot\) in front of the whole thing, you do not negate each piece separately

  25. princesspixie
    • 2 years ago
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    ok how do I explain the reasoning for it?

  26. satellite73
    • 2 years ago
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    you have some statement "blah blah blah" the the negation is "it is not true that blah blah blah"

  27. satellite73
    • 2 years ago
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    if you want to negate it you have to rewrite it as \[(\lnot p \lor q)\land (\lnot q\lor r)\] then negate it as \[\lnot \left((\lnot p \lor q)\land (\lnot q\lor r)\right)\]

  28. princesspixie
    • 2 years ago
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    so both should be included in the answer?

  29. satellite73
    • 2 years ago
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    yes but we have more work to do

  30. princesspixie
    • 2 years ago
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    Im still not sure how I should explain the answer ..

  31. satellite73
    • 2 years ago
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    unless you are unfamiliar with these laws. perhaps you are supposed to do something else

  32. satellite73
    • 2 years ago
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    your original statement All mathematicians must be good logicians and all good logicians must justify their claims is correct

  33. princesspixie
    • 2 years ago
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    ive never seen that little hook thing, but i see that in the textbook it says converse inverse and contrapostive .. not sure if that helps

  34. satellite73
    • 2 years ago
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    actually i would get rid of the word "must"

  35. satellite73
    • 2 years ago
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    "all mathematicians are good logicians and all good logicians justify their claims" perhaps is better

  36. princesspixie
    • 2 years ago
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    thats why i thought this symbol would be in the equation ~

  37. satellite73
    • 2 years ago
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    \(\lnot\) is the same as ~

  38. satellite73
    • 2 years ago
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    means "not"

  39. princesspixie
    • 2 years ago
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    ok so should i use the ~ if that what the text book uses?

  40. princesspixie
    • 2 years ago
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    ok .. how should i explain that in simple terms?

  41. princesspixie
    • 2 years ago
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    is this good... The symbol ~ means not. Therefore, if a person is not a good mathematician then the person is not a good logician, and the person cannot justify their claims.

  42. satellite73
    • 2 years ago
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    i would think perhaps the negation, without using the symbols, would be not all mathematicians are good logicians or not all logicians justify their claims

  43. princesspixie
    • 2 years ago
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    ok so i should use that instead of if a person is not a good... ?

  44. satellite73
    • 2 years ago
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    if a person is a good mathematician, then they are a good logician the statement if a person is not a good mathematician then the person is not a good logician is not the negation, it is the "inverse"

  45. princesspixie
    • 2 years ago
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    ok I understand

  46. princesspixie
    • 2 years ago
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    so would that be a good explanation?

  47. satellite73
    • 2 years ago
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    then negation of "if a person is a good mathematician, then they are a good logician" is there is some mathematician who is not a good logician

  48. princesspixie
    • 2 years ago
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    thats the answer for C?

  49. satellite73
    • 2 years ago
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    i forget what C is

  50. princesspixie
    • 2 years ago
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    c) Now use part b) to write the negation of this statement verbally.

  51. satellite73
    • 2 years ago
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    the negation of the whole thing would read something like "there is a good mathematician who is not a good logician, or there is a good logician who does not justify their claims

  52. satellite73
    • 2 years ago
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    the negation of the "and" statement is one or the other is false

  53. satellite73
    • 2 years ago
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    really to do this symbolically takes a while

  54. princesspixie
    • 2 years ago
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    ok I see

  55. satellite73
    • 2 years ago
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    \[\lnot \left((\lnot p \lor q)\land (\lnot q\lor r)\right)\] \[\lnot(\lnot p\lor q)\lor \lnot(\lnot q \lor r)\] \[(p\land \lnot q)\lor (q\land \lnot r)\]

  56. satellite73
    • 2 years ago
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    the last statement says there is a good mathematician who is not a good logician or there is a good logician who does not justify their claims

  57. princesspixie
    • 2 years ago
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    oh i see ok thanks!!

  58. satellite73
    • 2 years ago
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    yw good luck with this

  59. princesspixie
    • 2 years ago
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    thanks so much for all your help!!

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