Prove or given a counterexample: If and and function such that and , then f and g are both one-to-one and onto and .
To prove or provide a counterexample for the given statement.
Consider that are function such that then objective is to show that are both one-to-one and onto and .
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 said to be onto function if each element in co-domain is mapped with atleast one element in domain.
First show that is one-to-one, for this consider , where .
This implies that,
Since therefore, implies that .
This implies that
This shows that is one-to-one.
For this consider and objective is to show that there exists such that
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