Explain the main idea behind a proof by contrapositive and its main components.

Imagine you are writing a math textbook about Contradiction and  Contrapositive :
Proof by Contradiction
a.    Provide a brief write up about the main idea behind a proof by contradiction and its main components.
b.    Provide one example1 of a statement/theorem that can most naturally be proved using proof by contradiction.
c.    Provide an example1 of a statement/theorem that can be proved by both direct proof and proof by contradiction and discuss advantages and disadvantages of each approach.

4.    Proof by Contrapositive
a.    In a couple of sentences explain the main idea behind a proof by contrapositive and its main components.
b.    Provide an example1 of a statement/theorem that can be proved by both proof by contradiction and proving the contrapositive and discuss advantages and disadvantages of each approach.
c.    Provide an example1 of a statement/theorem that can be proved using contradiction but not by proving the contrapositive and discuss why that is.

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