Peirce’s 1870 “Logic Of Relatives” • Comment 11.10

In the case of a dyadic relation $F \subseteq X \times Y$ that has the qualifications of a function $f : X \to Y,$ there are a number of further differentia that arise:

$\begin{array}{lll} f ~\text{is surjective} & \iff & f ~\text{is total at}~ Y. \\[6pt] f ~\text{is injective} & \iff & f ~\text{is tubular at}~ Y. \\[6pt] f ~\text{is bijective} & \iff & f ~\text{is}~ 1\text{-regular at}~ Y. \end{array}$

For example, the function $f : X \to Y$ depicted below is neither total nor tubular at its codomain $Y,$ so it can enjoy none of the properties of being surjective, injective, or bijective.

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An easy way to extract a surjective function from any function is to reset its codomain to its range. For example, the range of the function $f$ above is $Y^\prime = \{ 0, 2, 5, 6, 7, 8, 9 \}.$ Thus, if we form a new function $g : X \to Y^\prime$ that looks just like $f$ on the domain $X$ but is assigned the codomain $Y^\prime,$ then $g$ is surjective, and is described as a mapping onto $Y^\prime.$

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The function $h : Y^\prime \to Y$ is injective.

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The function $m : X \to Y$ is bijective.

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