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| ===Commentary Note 11.10=== | | ===Commentary Note 11.10=== |
| | | |
− | <pre>
| + | In the case of a 2-adic relation ''F'' ⊆ ''X'' × ''Y'' that has the qualifications of a function ''f'' : ''X'' → ''Y'', there are a number of further differentia that arise: |
− | In the case of a 2-adic relation F c X x Y that has | |
− | the qualifications of a function f : X -> Y, there | |
− | are a number of further differentia that arise: | |
| | | |
− | | f is "surjective" iff f is total at Y. | + | <blockquote> |
− | | | + | {| cellpadding="4" |
− | | f is "injective" iff f is tubular at Y. | + | | ''f'' is "surjective" |
− | | | + | | iff |
− | | f is "bijective" iff f is 1-regular at Y. | + | | ''f'' is total at ''Y''. |
| + | |- |
| + | | ''f'' is "injective" |
| + | | iff |
| + | | ''f'' is tubular at ''Y''. |
| + | |- |
| + | | ''f'' is "bijective" |
| + | | iff |
| + | | ''f'' is 1-regular at ''Y''. |
| + | |} |
| + | </blockquote> |
| | | |
− | For example, or more precisely, contra example, | + | For example, or more precisely, contra example, the function ''f'' : ''X'' → ''Y'' that is depicted below is neither total at ''Y'' nor tubular at ''Y'', and so it cannot enjoy any of the properties of being sur-, or in-, or bi-jective. |
− | the function f : X -> Y that is depicted below | |
− | is neither total at Y nor tubular at Y, and so | |
− | it cannot enjoy any of the properties of being | |
− | sur-, or in-, or bi-jective. | |
| | | |
| + | <pre> |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| o o o o o o o o o o X | | o o o o o o o o o o X |
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| o o o o o o o o o o Y | | o o o o o o o o o o Y |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| + | </pre> |
| | | |
− | A cheap way of getting a surjective function out of any function | + | A cheap way of getting a surjective function out of any function is to reset its codomain to its range. For example, the range of the function ''f'' above is ''Y''′ = {0, 2, 5, 6, 7, 8, 9}. Thus, if we form a new function ''g'' : ''X'' → ''Y''′ that looks just like ''f'' on the domain ''X'' but is assigned the codomain ''Y''′, then ''g'' is surjective, and is described as mapping "onto" ''Y''′. |
− | is to reset its codomain to its range. For example, the range | |
− | of the function f above is Y'= {0, 2, 5, 6, 7, 8, 9}. Thus, | |
− | if we form a new function g : X -> Y' that looks just like | |
− | f on the domain X but is assigned the codomain Y', then | |
− | g is surjective, and is described as mapping "onto" Y'. | |
| | | |
| + | <pre> |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| o o o o o o o o o o X | | o o o o o o o o o o X |
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| o o o o o o o Y' | | o o o o o o o Y' |
| 0 2 5 6 7 8 9 | | 0 2 5 6 7 8 9 |
| + | </pre> |
| | | |
− | The function h : Y' -> Y is injective. | + | The function ''h'' : ''Y''′ → ''Y'' is injective. |
| | | |
| + | <pre> |
| 0 2 5 6 7 8 9 | | 0 2 5 6 7 8 9 |
| o o o o o o o Y' | | o o o o o o o Y' |
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| o o o o o o o o o o Y | | o o o o o o o o o o Y |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| + | </pre> |
| | | |
− | The function m : X -> Y is bijective. | + | The function ''m'' : ''X'' → ''Y'' is bijective. |
| | | |
| + | <pre> |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| o o o o o o o o o o X | | o o o o o o o o o o X |