P2.9.7 Let A be a set and f a bijection from A to itself. We say...

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P2.9.7 Let A be a set and f a bijection from A to itself. We say that f fixes an element r of A if f(x) = r. (a) Write a quan

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The required quantified statement is "For $f$ there exist a unique $yE A$ such that $Уr €A$ with $x\neq y$ , $f$ fixes $x$ ". Here y is that exactly one element which doesn't get fixed by f.

(b). Let the statement in (a) holds and A has more than one element. So, $f$ does not fix $y$ , then $f(y)\neq y$ . Let $f(y)=y_1$ (say). Clearly $y\neq y_1$ as $f(y)$ . Now as $f$ is a bijection, $y\neq y_1$ implies $f(y_1)\neq f(y)=y_1$ . So $f$ does not fix $y_1$ also.

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