Two-sided inverse

From CS2800 wiki
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]