Suppose and are both one-to-one and onto. Prove that exists and that .
If are both one-to-one and onto functions, then exist and
Suppose are both one-to-one and onto.
A function is said to be one-to-one function if the distinct elements in domain must be mapped with distinct elements in co-domain.
A function is onto function if each element in co-domain is mapped with atleast one element in domain.
Consider the one-to-one and onto functions
As the functions are one-to-one so the composite function is one-to-one.
As the functions are onto so the composite function is onto.
Thus the composite function is one-to-one and onto.
As the function is one-to-one correspondence, so there must exist inverse for the function from ,
Find the inverse of the composite function
This is true for all , thus
Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!Get Started