# Definiton of function

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 $f(a)=b$

• Most people consider the combination of both of them to be the definition of a function. – vadim123 Dec 16 '14 at 22:36
• so they are both true? strictly speaking, I think only combination of them is true. If they are stated separately, true or wrong? – Evan Dec 16 '14 at 22:39
• Both are true.. – Peter Franek Dec 16 '14 at 22:39
• As far as I observe that, those are equivalent to the definition of a function. – Luke Dec 16 '14 at 22:41

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 $(x,y)\in f$).