How to show 2 functions are an isomorphism
- Thread starter Cratylus
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Dr.Peterson
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Show that each of them is an isomorphism!
I think you'll need to be more specific. Do you have an example? Can you at least say what kind of objects you are working with?
I think you'll need to be more specific. Do you have an example? Can you at least say what kind of objects you are working with?
Cratylus
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I thought of that
Let E and F be partially ordered classes, and let g : E → F be an isomorphism. Prove that for arbitrary x ∈ E, g(Sx)=Sg(x)
conclude that Sx [MATH]\cong[/MATH] Sg(x).
from A book of set theory by Pinter
4.15 Definition If A and B are partially ordered classes and there exists an isomorphism from A to B, we say that A is isomorphic with B
Once showing them are isomorphisms deduce they are equal to each other.
G is injective: g(x)=g(y) => Sx [MATH] \cong [/MATH]Sy Suppose to the contrary x [MATH]\ne[/MATH]y, then x <y but by Def 4.15 ,this is a contradiction. Hence x=y So g is injective
Let E and F be partially ordered classes, and let g : E → F be an isomorphism. Prove that for arbitrary x ∈ E, g(Sx)=Sg(x)
conclude that Sx [MATH]\cong[/MATH] Sg(x).
from A book of set theory by Pinter
4.15 Definition If A and B are partially ordered classes and there exists an isomorphism from A to B, we say that A is isomorphic with B
Once showing them are isomorphisms deduce they are equal to each other.
G is injective: g(x)=g(y) => Sx [MATH] \cong [/MATH]Sy Suppose to the contrary x [MATH]\ne[/MATH]y, then x <y but by Def 4.15 ,this is a contradiction. Hence x=y So g is injective