SP20:Lecture 7 prep

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Due to a computer malfunction, I didn't have prep notes up for this lecture. However, the definitions we discussed are:

Definition: Identity function
If [math]A [/math] is a set, then the identity function on [math]A [/math] (written [math]\href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id}_A [/math] or simply [math]\href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id} [/math] if [math]A [/math] is clear) is the function [math]\href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id} : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} A [/math] given by [math]id(x)\href{/cs2800/wiki/index.php/Definition}{:=x} [/math].


Definition: Composition
Given a function [math]f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} B [/math] and a function [math]g : B \href{/cs2800/wiki/index.php/%E2%86%92}{→} C [/math], the composition of [math]g [/math] with [math]f [/math] (written [math]g \href{/cs2800/wiki/index.php/%E2%88%98}{∘} f [/math]) is the function [math]g \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} C [/math] given by [math](g \href{/cs2800/wiki/index.php/%5Ccirc}{\circ} f)(a) := g(f(a)) [/math].
Definition: Left inverse
Given a function [math]f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} B [/math], a left inverse [math]g [/math] of [math]f [/math] is a function [math]g : B \href{/cs2800/wiki/index.php/%E2%86%92}{→} A [/math] satisfying [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} [/math].
Definition: Right inverse
Given a function [math]f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} B [/math], a right inverse [math]g [/math] of [math]f [/math] is a function [math]g : B \href{/cs2800/wiki/index.php/%E2%86%92}{→} A [/math] satisfying [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} [/math].
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].