Difference between revisions of "Equivalence relation induced by a function"
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{{Requires proof|grade=D|easy=true|msg=Easy proof, marked as such. Just gotta show it's an [[equivalence relation]]}} | {{Requires proof|grade=D|easy=true|msg=Easy proof, marked as such. Just gotta show it's an [[equivalence relation]]}} | ||
==See also== | ==See also== | ||
| − | * [[ | + | * [[Factoring a function through the projection of an equivalence relation induced by that function yields an injection]] |
| − | {{Todo|Link to continuous version}} | + | ** [[If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection]] |
| + | {{Todo|Link to continuous version ([[File:MondTop2016ex1.pdf]] - Q5)}} | ||
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==References== | ==References== | ||
<references/> | <references/> | ||
{{Relations navbox|plain}} | {{Relations navbox|plain}} | ||
{{Definition|Elementary Set Theory|Set Theory}} | {{Definition|Elementary Set Theory|Set Theory}} | ||
Revision as of 22:28, 8 October 2016
Stub grade: B
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Something weird happened with every surjective map gives rise to an equivalence relation this page is what it SHOULD be. I also have a reference, granted not that strong of one
Grade: A
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Homework assignment isn't sufficient
Contents
Statement
Let [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] be sets and let [ilmath]f:X\rightarrow Y[/ilmath] be any mapping between them. Then [ilmath]f[/ilmath] induces an equivalence relation, [ilmath]\sim\subseteq X\times X[/ilmath] where[1]:
- for [ilmath]x_1,x_2\in X[/ilmath] we say [ilmath]x_1\sim x_2[/ilmath] if [ilmath]f(x_1)=f(x_2)[/ilmath]
Note that [ilmath]f[/ilmath] may be factored through the canonical projection of an equivalence relation to yield an injection. Furthermore if [ilmath]f[/ilmath] is surjective, then so is the induced map, and then the induced map is a bijection.
- See: factoring a function through the projection of an equivalence relation induced by that function yields an injection and if a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection
Proof
Grade: D
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
This proof has been marked as an page requiring an easy proof
The message provided is:
Easy proof, marked as such. Just gotta show it's an equivalence relation
This proof has been marked as an page requiring an easy proof
See also
TODO: Link to continuous version (File:MondTop2016ex1.pdf - Q5)
References
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