Suppose f:X→Y and g:Y→Z and both of these are one-to-one and onto. Prove that (g∘f)^(-1) exists and that (g∘f)^(-1)=f^(-1)∘g^(-1).
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Let
both are one-one and onto functions.
It's mean defined.
(1)Prove that
exist:-
we know that if both functions are ono-one , onto and gof defined,
then defined and exist too.
such that
(2)prove that
:-
now
is one-one onto
is one-one onto
so domain and codomain of and are same.
let
such that
and
............(1)
again
........from equation (3)
thus
from eq (1) and the last eq,