Difference between revisions of "Involution"
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| − | An involution is a function whose inverse is itself. | + | An involution is a function whose inverse is itself. That is, <math>f(f(x))=x</math>. |
| + | From the perspective of set theory and functions, if a relation is a function and is symmetric, then it is an involution. | ||
== Examples == | == 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 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 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 | + | * 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. | ||
== Properties == | == Properties == | ||
| − | * | + | * Function <math>f:X \rightarrow X</math> is an involution <math>\iff</math> <math>\forall x \in X</math> <math>f(x) = y \land f(y) = x</math>. This induces that both <math>(x,y)</math> and <math>(y,x)</math> are in f. By the definition of the inverse of a function, <math>\{ (y,x) | (x,y) \in f \}</math> is the inverse of the function f. Therefore, the function f must contain <math>f^{-1}</math>. From this, it is obtained that <math>f^{-1} \subseteq f</math>. Simmilalry, we can show that <math>f \subseteq f^{-1}</math>. Thus, <math>f = f^{-1}</math>. |
| + | Rewriting the first line we have: function <math>f:X \rightarrow X</math> is an involution <math>\iff</math> <math>f = f^{-1}</math>. | ||
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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. | ||
| + | * All involutions are [[bijection|bijections]]. | ||
| + | {{stub}} | ||
Latest revision as of 07:05, 9 May 2024
An involution is a function whose inverse is itself. That is, 
.
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
has the inverse 
, which is the same function, and thus 
is an involution. - The logical NOT is an involution because
. - The additive negation is an involution because
. - The identity function
is an involution because 
therefore, 
and 
. Hence, it is an involution. - The multiplicative inverse is an involution because
. In fact, for any 
is an involution.
has the inverse 
, which is the same function, and thus 
is an involution.
.
.
is an involution because 
therefore, 
and 
. Hence, it is an involution.
. In fact, for any 
is an involution.Properties
- Function
is an involution 
. This induces that both 
and 
are in f. By the definition of the inverse of a function, 
is the inverse of the function f. Therefore, the function f must contain 
. From this, it is obtained that 
. Simmilalry, we can show that 
. Thus, 
.
. This induces that both 
and 
are in f. By the definition of the inverse of a function, 
is the inverse of the function f. Therefore, the function f must contain 
. From this, it is obtained that 
. Simmilalry, we can show that 
. Thus, Rewriting the first line we have: function 
is an involution 
.
- A function is an involution iff it is symmetric about the line in the coordinate plane.
- All involutions are bijections.
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