Definitions using if instead of iff

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In summary: Bourbaki books, setting if to iff can be safe.In summary, in mathematical definitions, the use of "if" implies "if and only if" and allows for future refinement or generalization. This convention is used to avoid conflicts and arguments between different definitions of the same concept. However, in theorems or propositions, "if" only implies one direction of the conditional statement. This distinction is important to understand in order to correctly interpret and apply mathematical definitions and theorems.
  • #1
Tom1992
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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?
 
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  • #2
"iff", signifying "if and only if...", refers to a conditional statement and the converse of the conditional statement implying each other.

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.
 
  • #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.
 
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  • #4
symbolipoint said:
"iff", signifying "if and only if...", refers to a conditional statement and the converse of the conditional statement implying each other.

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.

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.
 
  • #5
Tom1992 said:
when a mathematical definition uses the word "if", can we assume that it also means "iff"?
Yes.I think I even remember "Bourbaki" making explicit mention of this "convention" in one of "his" books. :uhh:
 
  • #6
quasar987 said:
Yes.


I think I even remember "Bourbaki" making explicit mention of this "convention" in one of "his" books. :uhh:

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.
 
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  • #7
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 "<==".
 
  • #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.
 
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  • #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: "
 
  • #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.
 
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1. What is the difference between using "if" and "iff" in a definition?

Using "if" in a definition means that the statement is a sufficient condition for the defined term. In other words, if the statement is true, then the defined term must also be true. On the other hand, using "iff" in a definition means that the statement is both a sufficient and necessary condition for the defined term. This means that if the statement is true, then the defined term must also be true, and if the defined term is true, then the statement must also be true.

2. When should I use "if" instead of "iff" in a definition?

"If" should be used in a definition when the statement is only a sufficient condition for the defined term. This means that the statement being true is enough to guarantee the truth of the defined term. However, if the statement is also a necessary condition for the defined term, then "iff" should be used.

3. Can I use "if" and "iff" interchangeably in a definition?

No, "if" and "iff" have different meanings in a definition and should not be used interchangeably. Using the wrong term can change the meaning of the definition and lead to confusion.

4. Are there any advantages to using "if" instead of "iff" in a definition?

Using "if" in a definition can make the statement simpler and easier to understand. It also allows for more flexibility, as the statement may be true in some cases but not all cases. This can be advantageous in certain situations.

5. How do I determine whether to use "if" or "iff" in a definition?

Determining whether to use "if" or "iff" in a definition depends on the intended meaning of the statement and the defined term. If the statement is both a sufficient and necessary condition for the defined term, then "iff" should be used. If the statement is only a sufficient condition, then "if" should be used.

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