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If only I understood only if

  1. Apr 5, 2014 #1

    Atomised

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    1. Question in book is:

    State which of the following are true / false





    a) n = 3 only if n^2 - 2n - 3 = 0

    b) n^2 - 2n - 3 = 0 only if n=3

    c) If n^2 - 2n - 3 = 0 then n = 3

    The solutions it gives are

    a) True b) False c) False





    2. My assumptions

    P only if Q is logically equivalent to If Q then P





    3. The attempt at a solution

    Taking P to be n=3
    And Q to be n^2 - 2n - 3 = 0

    Restating the question

    a) Q implies P

    b) P implies Q

    c) Q implies P

    Since a) & c) are logically equivalent they must have the same answer yet the printed solution states otherwise.

    What am I not getting?


    Many thanks
     
  2. jcsd
  3. Apr 5, 2014 #2

    adjacent

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    a) If n=3 then n^2 - 2n - 3 = 0 - Yes if n is equal to 3 , then n^2- 2n - 3 is equal to zero
    b)If n^2 - 2n - 3 = 0 then n=3 - No. n can be -1 too.
    c)If n^2 - 2n - 3 = 0 then n = 3 - This means the same as b.
     
  4. Apr 5, 2014 #3

    Atomised

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    Thank you

    So X only if Y is logically equivalent to if X then Y?
     
  5. Apr 5, 2014 #4

    PeroK

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    Suppose: X only if Y:

    X True, Y True (Yes)
    X True, Y False (No)
    X False, Y True (Yes)
    X False, Y False (Yes)

    Suppose: If X, then Y:

    You can confirm that the above holds. So, yes they are the same.

    Note: I've used "yes" for this combination does not break the rule; and, "no" for breaks the rule.

    Note "only if" is really only used to test your logical thinking. Because of the above equivalence, in practice most people use "if X then Y".

    You can also check from the above table that these are also equivalent to "If not Y, then not X".
     
  6. Apr 5, 2014 #5

    Atomised

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  7. Apr 5, 2014 #6

    PeroK

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    It's a bit misleading because it doesn't say clearly what the "if" and "only if" apply to. The way to interpret that truth table is:

    P iff Q means "P if Q" and "P only if Q"; which is equivalent to "Q => P" and "P => Q"
     
  8. Apr 5, 2014 #7

    AlephZero

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    http://en.wikibooks.org/wiki/Mathematical_Proof/Introduction/Logical_Reasoning gives the wrong symbols under the heading "Implication types", and later it uses talks about "existence" instead of "truth". Statements like
    are at best very confusing IMO.

    I would treat it the same as the rest of Wikipedia, i.e. assume what is says is true only if you already know it is true :smile:

    For a better explanation, see http://www.math.csusb.edu/notes/logic/lognot/node1.html and http://www.math.csusb.edu/notes/logic/eequiv/eequiv.html
     
  9. Apr 5, 2014 #8

    "X if Y" means "if Y then X," clearly.

    I'll also assume that you know that "X if and only if Y" means "if X then Y, and if Y then X".

    Therefore we can conclude that "X only if Y" must be "if X then Y" because adding this to "X if Y" adds that implication to its logical meaning.
     
  10. Apr 5, 2014 #9

    Atomised

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    AlephZero - thanks for busting wikipedia - I should know better than to be misled by it.

    1MileCrash - I am now having the aha moment... of course the 'only if' is the other direction from 'if' in iff, also a brilliant way of remembering it thank you, job done.
     
  11. Apr 6, 2014 #10

    adjacent

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    :rofl: So true
     
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