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Formal Definition

A function $\phi:X \rightarrow Y$ is onto (or surjective) if $\forall y \in Y, \exists x \in X$ such that $f(x)=y$.

Informal Definition

A function is onto if every element of its codomain has some element of the function's domain that maps to it.


Say we have a function $\phi:X \rightarrow Y$ where $\phi(x) = 5x$. In order to show $\phi$ is onto, we must show $\exists x \in X$ such that $\phi(x) = y$. Let $x=(1/5) y$. Then, $\phi(x)=5[(1/5) y]=y$. Therefore, $\phi$ is onto. $\blacksquare$


Say we have a function $\phi:\mathbb{Z} \rightarrow \mathbb{R}$ where $\phi(x) = x$ ($x \in \mathbb{Z}$ and $y \in \mathbb{R}$). Let $y = 1.5$. Since the domain of $\phi$ is only the integers, it is obvious that $\nexists x$ such that $\phi(x) = 1.5 = y$. $\blacksquare$

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