Difference between revisions of "Two-sided inverse"

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{{Def|1=Two-sided inverse|2=
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<noinclude>[[Category:Functions]] [[Category:Definition]]</noinclude>{{Definition:Two-sided inverse}}
If <math>f : A [[→]] B</math> is a [[function]], then <math>g : B [[]] A</math>
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is a [[two-sided inverse]] of <math>f</math> (written <math>g = f^{[[-1]]}</math>)
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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]].
if <math>f [[\circ]] g [[Equality (functions)|=]] [[id]]_B</math> and
 
<math>g [[\circ]] f [[Equality (functions)|=]] [[id]]_A</math>.
 
}}
 
  
 
<noinclude>{{:Example:inverses}}</noinclude>
 
<noinclude>{{:Example:inverses}}</noinclude>

Latest revision as of 13:00, 5 February 2020

Definition: Two-sided inverse
If [math]f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} B [/math] is a function, then [math]g : B \href{/cs2800/wiki/index.php/%E2%86%92}{→} A [/math] is a two-sided inverse of [math]f [/math] if [math]f \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} g \href{/cs2800/wiki/index.php/Equality_(functions)}{=} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id}_B [/math] and [math]g \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} f \href{/cs2800/wiki/index.php/Equality_(functions)}{=} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id}_A [/math].

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 [math]f^{\href{/cs2800/wiki/index.php/-1}{-1}} [/math] to denote the inverse of [math]f [/math] if it exists.


If [math]f \href{/cs2800/wiki/index.php/Definition}{:=} [/math] Fun-abc-12-a1b2c2.svg and [math]g \href{/cs2800/wiki/index.php/Definition}{:=} [/math] Fun-12-abc-1a2b.svg then [math]f [/math] is a left inverse of [math]g [/math], because

[math]f \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} g \href{/cs2800/wiki/index.php/Equality_(functions)}{=} [/math] Fun-id-12.svg [math]\href{/cs2800/wiki/index.php/Equality_(functions)}{=} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id} [/math]

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

[math]g \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} f \href{/cs2800/wiki/index.php/Equality_(functions)}{=} [/math] Fun-abc-abc-aabbcb.svg [math]\href{/cs2800/wiki/index.php/Equality_(functions)}{\neq} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id} [/math]

Finally, if [math]h := [/math] Fun-abc-123-a2b3c1.svg and [math]i := [/math] Fun-123-abc-1c2a3b.svg, then [math]h [/math] and [math]i [/math] are two-sided inverses of each other, because

[math]h \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} i \href{/cs2800/wiki/index.php/Equality_(functions)}{=} [/math] Fun-id-123.svg [math]\href{/cs2800/wiki/index.php/Equality_(functions)}{=} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id} [/math] and [math]i \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} h \href{/cs2800/wiki/index.php/Equality_(functions)}{=} [/math] Fun-id-abc.svg [math]\href{/cs2800/wiki/index.php/Equality_(functions)}{=} \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id} [/math]