Let X be a metric space and f:X→X be such that f(f(x))=x, for all x∈X.
Then f
- is one-one and onto;
- is one-one but not onto;
- is onto but not one-one;
- 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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