## 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.

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.

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

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 $Y^\prime.$

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

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

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

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

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

### Resources

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