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Difference between revisions of "Involution"
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An involution is a function whose inverse is itself.  
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An involution is a function whose inverse is itself.
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From the perspective of set theory and functions, if a relation is a function and is symmetric, then it is an involution.
  
  
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* The [[logical NOT]] is an involution because <math>\neg { \neg p} \equiv p</math>.
 
* The [[logical NOT]] is an involution because <math>\neg { \neg p} \equiv p</math>.
 
* The additive negation is an involution because <math>--x=x</math>.
 
* The additive negation is an involution because <math>--x=x</math>.
 +
* The identity function <math>I_x</math> is an involution because <math>I_x:X \rightarrow X = \{(x,x) | x \in X\}</math> therefore, <math>\forall (x,x) \in I_x</math> <math>f(x) =  x</math> and <math>f(f(x)) = x</math>. Hence, it is an involution.
 
* The [[multiplicative inverse]] is an involution because <math>\frac{1}{\frac{1}{x}}=x</math>. In fact, for any <math>n \neq 0, f(x)=\frac{n}{x}</math> is an involution.
 
* The [[multiplicative inverse]] is an involution because <math>\frac{1}{\frac{1}{x}}=x</math>. In fact, for any <math>n \neq 0, f(x)=\frac{n}{x}</math> is an involution.
  
 
== Properties ==
 
== Properties ==
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* An function is an involution [[iff]] it is symmetric about the line <math>f(x)=x</math> in the coordinate plane.
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* A function is an involution [[iff]] it is symmetric about the line <math>f(x)=x</math> in the coordinate plane.
  
 
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Revision as of 16:17, 8 September 2021

An involution is a function whose inverse is itself. From the perspective of set theory and functions, if a relation is a function and is symmetric, then it is an involution.


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 $\neg { \neg p} \equiv p$.
  • The additive negation is an involution because $--x=x$.
  • The identity function $I_x$ is an involution because $I_x:X \rightarrow X = \{(x,x) | x \in X\}$ therefore, $\forall (x,x) \in I_x$ $f(x) = x$ and $f(f(x)) = x$. Hence, it is an involution.
  • 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

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

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