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If f maps Z on to Z is an isomorphism, prove that f is the identity map.
Solution
f:Z->Z is an isomorphism
let ,x Z and y Z (codomain) ; = belongs to
hence, y=f(x) ;y is as function of x
isomorphism means it is onto and one-one
onto:
"every" yZ (codomain) ,there exist xZ ,such that f(x)=y ...1
one-one:
for "any" yZ(codomain) ,there exist at most one xZ such that y=f(x) ...2
from 1 and 2, we get "every" yZ(codomain) has at most one xZ such that y=f(x) and f:Z->Z.
which means every element of set Z maps to the same element in Z(codomain), since domain
and range are same i.e Z=Z. hence identity map.

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pIf f maps Z on to Z is an isomorphism, prove that f is the identi.pdf

  • 1. If f maps Z on to Z is an isomorphism, prove that f is the identity map. Solution f:Z->Z is an isomorphism let ,x Z and y Z (codomain) ; = belongs to hence, y=f(x) ;y is as function of x isomorphism means it is onto and one-one onto: "every" yZ (codomain) ,there exist xZ ,such that f(x)=y ...1 one-one: for "any" yZ(codomain) ,there exist at most one xZ such that y=f(x) ...2 from 1 and 2, we get "every" yZ(codomain) has at most one xZ such that y=f(x) and f:Z->Z. which means every element of set Z maps to the same element in Z(codomain), since domain and range are same i.e Z=Z. hence identity map.