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If, only if, if and only if

  1. Sep 13, 2008 #1
    1. The problem statement, all variables and given/known data

    Insert either "if", "only if", or "if and only if"..

    x>2 ... x^2>9

    2. Relevant equations

    3. The attempt at a solution

    I dont think any fit :S coonffuuuuuused
  2. jcsd
  3. Sep 13, 2008 #2
    x can be any real number? In that case I *think* it should be "if".
    But I'm no expert at this :P
  4. Sep 13, 2008 #3
    if equals =>
    only if equals <=
    if and only if equals <=>
  5. Sep 13, 2008 #4
    One fits perfectly. Think about what dirk said.
  6. Sep 13, 2008 #5
    Oops sorry, guess I was wrong.
  7. Sep 13, 2008 #6
    still dont know :S

    I know the x^2>9 is equivalent to x<-3 or x>3.

    so im guessing that rules out the <=>

    and if it was the x>3, then I would stick the <= one in. ("only if").

    "if" obviously wont work.

    but dont you have to consider the x<-3 case too? or am I imposing somekind of weird necessary/sufficient crap in where i shouldn't?
  8. Sep 13, 2008 #7
    Ponder the following. First, assume that x>2 is true. Does this imply that x^2>9 is true? If yes, then "if x>2, then x^2>9". If no, then not "if x>2, then x^2>9". (This is the "sufficient" condition.) Second, assume that x^2>9 is true. Does this imply that x>2 is true? If yes, then "only if x>2, then x^2>9". If no, then not "only if x>2, then x^2>9". (This is the "necessary" condition.) If yes to both, then "if and only if".
  9. Sep 13, 2008 #8


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    Don't you also need to insert 'then' somewhere? :smile:
  10. Sep 13, 2008 #9
    I think you need to know if there are any constraints on x. Can it be only a positive integer, perhaps? What does it say in your book?
  11. Sep 14, 2008 #10
    thanks for the clarification. I kind just used my intuition before but glad to know how to properly do it, lol :).

    the answer is no and no then, because there isnt a constaint on x... :S

    so was I right in saying none?
  12. Sep 14, 2008 #11
    no specified constaints
  13. Sep 14, 2008 #12
    Yeah, if there are no constraints on x, then it seems to me that you were correct in that none fit. It is not necessary that x>2 for x^2>9. What's necessary is that x is not equal to 2. Neither is x>2 sufficient for x^2>9. (In other words I was wrong when I said one fits perfectly.)
    Last edited: Sep 14, 2008
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