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Problem in understanding the meaning

  1. Jun 25, 2013 #1
    What is the difference between the 2 expressions

    1) for all x belongs to ℝ there exists y belongs to ℝ such that f(x)=y
    2) there exists y belongs to ℝ such that for all x belongs to ℝ , f(x)=y

    I want to know the exact difference.
  2. jcsd
  3. Jun 25, 2013 #2


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    hi thudda! welcome to pf! :wink:

    tell us what you think, and then we'll comment! :smile:
  4. Jun 25, 2013 #3

    I think the 1st expression implies that for every x value there is a corresponding y value.And the 2nd imply that for all x values there's one or a set of y values...what I want to know is whether it is one y value or a set of y values..
  5. Jun 25, 2013 #4


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    hi thudda! :smile:
    yup … basically, it doesn't say anything more than that f is a function! :wink:
    ah, it's one y value …

    "there exists y" always means there exists a y :smile:

    (btw, when i see ##y\in R##, i always read that as "y in R" … it's shorter, and i think, easier, than "y belongs to R" :wink:)
  6. Jun 25, 2013 #5


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    Without more, both "there exists a(n)" and its abbreviation [itex]\exists[/itex] mean "there exists at least one". If you want to specify uniqueness, you must do so expressly. The abbreviation for "there exists exactly one" is [itex]\exists ![/itex] - the usual symbol followed by an exclamation mark.

    As to the OP's examples: The first isn't quite the definition of a function; it would be if it asserted that y was unique. In the second, it follows from the definition of a function that if such a y exists then it must be unique.
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