Friday, April 26, 2019

functions - f:XrightarrowX such that f(f(x))=x


Let X be a metric space and f:XX be such that f(f(x))=x, for all xX.



Then f




  1. is one-one and onto;

  2. is one-one but not onto;


  3. is onto but not one-one;

  4. need not be either.




From the given condition I have that f2=i where i is the identity function. If f itself is the identity function then the conditions are satisfied as well as f is bijection. Is that the only such function or are there other possibilities ?



My guess is that it will be bijection i.e. option 1 will be correct .



For see, if f(x1)=y and f(x2)=y then f(y)=x1 and f(y)=x2

will be possible iff x1=x2. So this is injective.




Now an injection from a set to itself is trivially surjective so it is bijective. Is my proof correct?

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