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Are these two statements true about the definition of a function $f$ from $A$ into $B$

  1. For every element $a \in A$, there exists at least one element $b \in B$ such that $f(a)=b$

  2. For every element $a \in A$, there exists at most one element $b \in B$ such that $f(a)=b$

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    $\begingroup$ Most people consider the combination of both of them to be the definition of a function. $\endgroup$ – vadim123 Dec 16 '14 at 22:36
  • $\begingroup$ so they are both true? strictly speaking, I think only combination of them is true. If they are stated separately, true or wrong? $\endgroup$ – Evan Dec 16 '14 at 22:39
  • $\begingroup$ Both are true.. $\endgroup$ – Peter Franek Dec 16 '14 at 22:39
  • $\begingroup$ As far as I observe that, those are equivalent to the definition of a function. $\endgroup$ – Luke Dec 16 '14 at 22:41
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The standard (Bourbaki) definition (which is said to be canonical) of a function from a set $X$ to $Y$ is: a function is $(f,X,Y)$ with $f\subset X\times Y$ such that for every $x\in X$ there is a unique $y\in Y$ with $(x,y)\in f$ (it is common to write $y=f(x)$ for the statement $(x,y)\in f$).

Thus both 1. and 2. are correct ("unique"="there is at least one and at most one").

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