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.

LOR 1870 Figure 40

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.

LOR 1870 Figure 41

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

LOR 1870 Figure 42

The function m : X \to Y is bijective.

LOR 1870 Figure 43
This entry was posted in Graph Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , . Bookmark the permalink.

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

  1. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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