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Involution

Revision as of 18:29, 19 September 2008 by Herefishyfishy1 (talk | contribs)

An involution is a function whose inverse is itself.


Examples

  • The function $y(x)=x$ has the inverse $x(y)=y$, which is the same function, and thus $f(x)=x$ is an involution.
  • The logical NOT is an involution because $\neg \neg p} \equiv p$ (Error compiling LaTeX. Unknown error_msg).
  • The additive negation is an involution because $--x=x$.
  • The multiplicative inverse is an involution because $\frac{1}{\frac{1}{x}}=x$. In fact, for any $n \neq 0$, $f(x)=\frac{n}{x}$ is an involution.

Properties

  • An function is an involution iff it is symmetric about the line $f(x)=x$ in the coordinate plane.
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