Difference between revisions of "Involution"

Revision as of 11:42, 17 January 2017

An involution is a function whose inverse is itself.

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 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.

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

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 == Examples ==
 * The function <math>y(x)=x</math> has the inverse <math>x(y)=y</math>, which is the same function, and thus <math>f(x)=x</math> is an involution.
-* 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 [[multiplicative inverse]] is an involution because <math>\frac{1}{\frac{1}{x}}=x</math>. In fact, for any <math>n \neq 0</math>, 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 ==
-* An function is an involution [[iff]] it is symmetric about the line </math>f(x)=x$ in the coordinate plane.
+* 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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