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$\mathrm{\left[}\mathrm{\right]}$ A function $\mathit{f}$ : $\mathrm{X}\mathrm{\to }\mathrm{Y}$ takes elements from X as the input and provides elements from $\mathrm{Y}$ as the output. It is important to note that

$\mathrm{•}$ $\mathit{f}$ produces only one element from $\mathrm{Y}$ as the output for any $\mathit{x}\mathrm{\in }\mathrm{X}\mathrm{.}$

$\mathrm{•}$ It must produce an output for every $\mathit{x}\mathrm{\in }\mathrm{X}$ otherwise it does not qualify as a function.

$\mathrm{•}$ $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\in }\mathrm{Y}$ is called the image of $\mathit{x}\mathrm{\in }\mathrm{X}\mathrm{.}$

$\mathrm{•}$ $\mathit{x}\mathrm{\in }\mathrm{X}$ is called the pre‐image of $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\in }\mathrm{Y}\mathrm{.}$

$\mathrm{•}$ We may thus say that for each $\mathit{x}\mathrm{\in }\mathrm{X}$ the function produces a unique image $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\in }\mathrm{Y}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ A self‐map $\mathit{f}$ : (X, ${\mathit{d}}_{\mathit{X}}$) $\mathrm{\to }\mathrm{\left(}\mathrm{X}\mathrm{,}\mathrm{ }{\mathit{d}}_{\mathit{X}}\mathrm{\right)}$ is a contraction if

$\mathrm{•}$ $\mathrm{\exists }\mathit{K}\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ such that

$\mathrm{•}\mathit{d}\mathrm{\left(}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{,}\mathrm{ }\mathit{f}\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{\right)}\mathrm{\le }\mathit{K}\mathit{d}\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{ }\mathit{y}\mathrm{\right)}$

$\mathrm{•}\mathrm{\forall }\mathit{x}\mathrm{,}$ $\mathit{y}\mathrm{\in }\mathrm{X}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ Given a non‐empty set X, let $\overline{\mathit{P}\mathrm{\left(}\mathrm{X}\mathrm{\right)}}$ denote the collection of all non‐empty subsets of X. $\mathrm{\left[}\mathrm{\right]}$ A correspondence ${\mathit{f}}^{\mathit{c}}$ : $\mathrm{X}\mathrm{\to }\overline{\mathit{P}\mathrm{\left(}\mathrm{Y}\mathrm{\right)}}$ takes elements from X as the input and provides elements from $\overline{\mathit{P}\mathrm{\left(}\mathrm{Y}\mathrm{\right)}}$ as the output. It is important to note that

$\mathrm{•}$ ${\mathit{f}}^{\mathit{c}}$ produces only one element from $\overline{\mathit{P}\mathrm{\left(}\mathrm{Y}\mathrm{\right)}}$ as the output for any $\mathit{x}\mathrm{\in }\mathrm{X}$; but one element of $\overline{\mathit{P}\mathrm{\left(}\mathrm{Y}\mathrm{\right)}}$ may contain one or more elements of Y.

$\mathrm{•}$ It must produce an output for every $\mathit{x}\mathrm{\in }\mathrm{X}$ otherwise it does not qualify as a corre‐ spondence.

$\mathrm{•}$ ${\mathit{f}}^{\mathit{c}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\in }\overline{\mathit{P}\mathrm{\left(}\mathrm{Y}\mathrm{\right)}}$ is called the image of $\mathit{x}\mathrm{\in }\mathrm{X}\mathrm{.}$

$\mathrm{•}$ $\mathit{x}\mathrm{\in }\mathrm{X}$ is called the pre‐image of ${\mathit{f}}^{\mathit{c}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\in }\overline{\mathit{P}\mathrm{\left(}\mathrm{Y}\mathrm{\right)}}\mathrm{.}$

$\mathrm{•}$ We will denote ${\mathit{f}}^{\mathit{c}}\mathrm{:}\mathrm{X}\mathrm{\to }\overline{\mathit{P}\mathrm{\left(}\mathrm{Y}\mathrm{\right)}}$ as ${\mathit{f}}^{\mathit{c}}\mathrm{:}\mathrm{X}\mathrm{=}\mathrm{Y}$ throughout this section.

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