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We often see the word "if" in connection with definitions e.g

y is a solution of $x^{'}=f(x,t)$ if $y^{'}=f(y,t)$ or

A metric space is complete if every cauchy sequence converges.

Why isnt it written if and only if in general in this context?

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marked as duplicate by Cameron Buie, Community Jul 15 '18 at 15:46

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  • $\begingroup$ Normally "iff" means "if and only if" in mathematical texts. $\endgroup$ – Cornman Jul 15 '18 at 15:40
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    $\begingroup$ It's conventional to use "if" instead of "if and only if" in definitions. You are correct in that "if and only if" is intended, but this is generally understood. $\endgroup$ – saulspatz Jul 15 '18 at 15:40
  • $\begingroup$ "A if B" means $B \Rightarrow A$. "A iff B" means $B \iff A$. $\endgroup$ – mvw Jul 15 '18 at 16:06
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    $\begingroup$ Because it is a definition. A definition is assumed to be declaring two expressions to be the same thing. The "if" in a definition is not mathematical but colloquial. We can replace it with "when" or "is defined to be" or even ":" if we want to. $\endgroup$ – fleablood Jul 15 '18 at 16:17

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