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?


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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