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[] A function f : XY takes elements from X as the input and provides elements from Y as the output. It is important to note that

f produces only one element from Y as the output for any xX.

It must produce an output for every xX otherwise it does not qualify as a function.

f(x)Y is called the image of xX.

xX is called the pre‐image of f(x)Y.

We may thus say that for each xX the function produces a unique image f(x)Y.

[] A self‐map f : (X, dX) (X,dX) is a contraction if

K(0,1) such that

d(f(x),f(y))Kd(x,y)

x, yX.

[] Given a non‐empty set X, let P(X)¯ denote the collection of all non‐empty subsets of X. [] A correspondence fc : XP(Y)¯ takes elements from X as the input and provides elements from P(Y)¯ as the output. It is important to note that

fc produces only one element from P(Y)¯ as the output for any xX; but one element of P(Y)¯ may contain one or more elements of Y.

It must produce an output for every xX otherwise it does not qualify as a corre‐ spondence.

fc(x)P(Y)¯ is called the image of xX.

xX is called the pre‐image of fc(x)P(Y)¯.

We will denote fc:XP(Y)¯ as fc:X=Y throughout this section.

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