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Definitions using if instead of iff

  1. Jan 26, 2007 #1
    definitions using "if" instead of "iff"

    when a mathematical definition uses the word "if", can we assume that it also means "iff"?

    for example, here's a defintion straight from a book:
    definition: a bijection f is a homeomorphism if f and its inverse are continuous.

    so this definition means that

    (f and its inverse are continuous) implies (f is a homeomorphism).

    but the converse is not stated because the definition uses "if" instead of "iff". So literally, based on the above definition, if f is a homeomorphism, then we cannot conclude that f and its inverse are continuous. but that's not true. what's going on? does "if" mean "iff" when we see "if" in a definition?
    Last edited: Jan 26, 2007
  2. jcsd
  3. Jan 26, 2007 #2


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    "iff", signifying "if and only if...", refers to a conditional statement and the converse of the conditional statement implying eachother.

    IF p THEN q AND IF q THEN p
    means the same as
    p IF AND ONLY IF q
    means the same as
    q IF AND ONLY IF p
    means the same as
    p is true whenever q is true and q is true whenever p is true.
  4. Jan 26, 2007 #3
    yes, i know what iff means. i'm not asking for the meaning of iff.

    what i'm asking is that if a definition uses "if" instead of "iff", does the definition mean "if" only, but not "if and only if"? or does the definition imply iff? see my above example for what i'm talking about.
    Last edited: Jan 26, 2007
  5. Jan 26, 2007 #4
    This was not the orignal poster's question, I believe he knows fairly well what if and iff mean. His question was as to whether in a definition the two are interchangeable, to which I believe the answer is yes.
  6. Jan 26, 2007 #5


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    I think I even remember "Bourbaki" making explicit mention of this "convention" in one of "his" books. :uhh:
  7. Jan 26, 2007 #6
    thanks for clarifying this to me. i wonder if there are many definitions out there that actually do mean "if" but not "iff". i hope not, because i will assume "if" means "iff" in every defintion i read from now on. i will also henceforth assume "if" means "iff" for definitions made within a proof and within a question as well.

    just now i was reading a problem that read "define aRb if a-b is rational. prove that..." holy crap, i wasted about 15 minutes trying to come up with the proof without success because i thought the "if" was to be read one way only (in which case i don't think there is a solution). now that i know that the "if" in the question means "iff", the solution is easy.
    Last edited: Jan 26, 2007
  8. Jan 26, 2007 #7


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    When you think about it, it would not make sense that the "if" in a definition did not mean "iff". Take your homeomorphism exemple for instance. If we take the "if" in the definition to really mean "if", then the sentence

    "Let f be a homeomorphism, [...]" bears no meaning because we can't say anything about f from the fact that it's a homeomorphism.

    The whole goal of a definition is to compactify writing by assigning a relatively short succession of words to a larger succession of words and symbols".

    We want to be able to say "Let f be an homeomorphism" instead of "Let f be continuous bijection whose inverse is continuous".

    So whenever a definition is made, it is to set, for the sake of simplicity, a logical equivalence (i.e. a iff relation) btw a succesion of worlds and symbols whole meaning is already known, to a new succession of symbols and words.

    Therefor, rest assured that no definition ever means "if" really as "<==".
  9. Jan 26, 2007 #8
    ok, i got it now. "if" in a definition (whether it be an official mathematical definition, a definition within a proof, or a definition within an exercise) means <==>, but "if" in a theorem means only one way.

    still trying to get over all this notation gobbledygook in math books.
    Last edited: Jan 26, 2007
  10. Jan 26, 2007 #9
    It is hard to disagree with the Bourbaki group, who I respect very much, but I think there is a reason that definitions use "if" rather than "iff".

    This has to do with generalization, consider:

    Definition 1: A function is integrable if the ....Reimann sums...

    Then in the next chapter:

    Definition 2: A function is integrable ...the sense of Lebesgue...

    Granted the example is imperfect, because we distinguish between these types of integrability by name, but I think this is why mathematicians make definitions using "if", to allow that the current definition is not a complete characterization but is open to future refinement/generalization.

    Even if this is never done, in spirit it is always good to leave the door open for others to develop your work. "I defined it and only I defined it:biggrin: "
  11. Jan 26, 2007 #10
    crosson has point. also mathematicians often make their own definitions of the same thing, so using iff in their personal definitions can lead to some arguments.
    Last edited: Jan 26, 2007
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