ONT Re: Logic Of Relatives
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LOR. Note 52
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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.
|
| f is "injective" iff f is tubular at Y.
|
| f is "bijective" iff f is 1-regular at Y.
For example, or more precisely, counterexample,
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.
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o o o o o o o o o o X
| \ | / \ \ | | \ /
| \ | / \ \ | | \ f
| \|/ \ \| | / \
o o o o o o o o o o Y
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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'.
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o o o o o o o o o o X
| \ | / \ \ | | \ /
| \ | / \ \ | | \ g
| \|/ \ \| | / \
o o o o o o o Y'
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The function h : Y' -> Y is injective.
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o o o o o o o Y'
| | \ / | \ /
| | \ | \ h
| | / \ | / \
o o o o o o o o o o Y
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The function m : X -> Y is bijective.
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o o o o o o o o o o X
| | | \ / \ / | \ /
| | | \ \ | \ m
| | | / \ / \ | / \
o o o o o o o o o o Y
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Jon Awbrey
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