Difference between revisions of "Two-sided inverse"

Revision as of 17:52, 18 February 2018

Definition: Two-sided inverse

If is a function, then is a two-sided inverse of if and .

If [math]f [/math] has a two-sided inverse, it must be unique, so we are justified in writing the two-sided inverse of [math]f [/math]. We also write to denote the inverse of [math]f [/math] if it exists.

If and then [math]f [/math] is a left inverse of [math]g [/math], because

For the same reason, [math]g [/math] is a right inverse of [math]f [/math].

However, [math]f [/math] is not a right inverse of [math]g [/math] (nor is [math]g [/math] a left inverse of [math]f [/math]) because

Finally, if [math]h := [/math] and [math]i := [/math] , then [math]h [/math] and [math]i [/math] are two-sided inverses of each other, because

and

@@ Line 1: / Line 1: @@
 {{Def|1=Two-sided inverse|2=
-If <math>f : A [[→]] B</math> is a [[function]], then <math>g : B [[→]] A</math>
+If <math>f : A [[→]] B</math> is a [[function]], then <math>g : B [[→]] A</math> is a [[two-sided inverse]] of <math>f</math> if <math>f [[\circ]] g [[Equality (functions)|=]] [[id]]_B</math> and <math>g [[\circ]] f [[Equality (functions)|=]] [[id]]_A</math>.
-is a [[two-sided inverse]] of <math>f</math> (written <math>g = f^{[[-1]]}</math>)
-if <math>f [[\circ]] g [[Equality (functions)|=]] [[id]]_B</math> and
-<math>g [[\circ]] f [[Equality (functions)|=]] [[id]]_A</math>.
 }}
+If <math>f</math> has a [[two-sided inverse]], [[Claim:two-sided inverses are unique|it must be unique]], so we are justified in writing ''the'' [[two-sided inverse]] of <math>f</math>.  We also write <math>f^{[[-1]]}</math> to denote the inverse of <math>f</math> if it [[exists]].
 <noinclude>{{:Example:inverses}}</noinclude>