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I often write my notes as logical statements and constantly wonder why people use only the "if" direction in the definitions instead of the "if and only if". Consider:

"A homomorphism $\phi$ is said to be an isomorphism if it is an injection and a surjection." would translate to:

$homomorphism(\phi) \wedge injection(\phi) \wedge surjection(\phi) \implies isomorphism(\phi)$

By adding the following statement consistent with the previous one:

$homomorphism(\phi) \wedge \neg injection(\phi) \wedge surjection(\phi) \implies isomorphism(\phi)$

we could deduce

$homomorphism(\phi) \wedge surjection(\phi) \implies isomorphism(\phi)$

which is obviously not what we intended to express in the definition, however the intention does not imply the deed.

From the strict logical perspective do you agree that all definitions of this form are "flawed" owing to the lack of the attention to details? Or is there some reason which I have not managed to observe that explains why "if" is used in lieu of "if and only if"? Would you accept if I write in papers definitions in the iff version? I.e. "A homomorphism $\phi$ is an isomorphism if and only if it is an injection and a surjection."

Dávid Natingga
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    To save chalk ? – Guest Dec 14 '13 at 22:20
  • The close votes reference three "duplicate" questions. I'll note that a site search for "iff definition" gives at least a couple more candidates. – Blue Dec 14 '13 at 22:38
  • I agree with you, but I think that your argument is valid only if you were in a position with all there is to know about the thing you are considering. . – CoffeeIsLife Dec 14 '13 at 22:41

2 Answers2

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That is a good question. It used to bother me too until I saw it in some undergraduate textbook say "following the standard convention of writing if instead of if and only if in definitions ... ", do not remember the exact wording but that is basically what the book said. Since that time this never bothered me, but whenever I teach classes I always make my definitions as if and only if statements.

Nicolas Bourbaki
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"If an only if" means "necessary and sufficient", which is a double implication.

"If" alone means"only if" or "sufficient", which is a single implication.

user115289
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