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

Peirce’s 1870 “Logic of Relatives”Comment 11.10

A dyadic relation F \subseteq X \times Y which qualifies as a function f : X \to Y may then enjoy a number of further distinctions.

Definitions

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

Function f : X → Y
\text{Figure 40. Function}~ f : X \to Y

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 \}.  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.

Function g : X → Y'
\text{Figure 41. Function}~ g : X \to Y'

The function h : Y' \to Y is injective.

Function h : Y' → Y
\text{Figure 42. Function}~ h : Y' \to Y

The function m : X \to Y is bijective.

Function m : X → Y
\text{Figure 43. Function}~ m : X \to Y

Resources

cc: CyberneticsOntolog ForumStructural ModelingSystems Science
cc: FB | Peirce MattersLaws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

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5 Responses to Peirce’s 1870 “Logic of Relatives” • Comment 11.10

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