Characterization of Pointwise Minimum of Real-Valued Functions

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Theorem

Let $S$ be a set.

Let $\R$ denote the real number line.

Let $f, g :S \to \R$ be real-valued functions.

Let $f \wedge g$ denote the pointwise maximum of $f$ and $g$, that is, $f \wedge g$ is the mapping defined by:

$\forall s \in S : \map {\paren{f \wedge g} } s = \min \set{\map f s, \map g s}$

Then:

$f \wedge g = \dfrac 1 2 \paren{f + g - \size{f - g}}$

where:

$f + g$ denotes the pointwise addition of $f$ and $g$

$f - g$ denotes the pointwise difference of $f$ and $g$

$\size{f - g}$ denotes the absolute value of $f - g$

$\dfrac 1 2 \paren{f + g - \size{f - g}}$ denotes the pointwise scalar multiplication of $f + g - \size{f - g}$ by $\dfrac 1 2$

Proof

We have:

\(\ds \forall s \in S: \, \)

\(\ds \map {\paren{f \wedge g} } s\)

\(=\)

\(\ds \min\set {\map f s, \map g s}\)

Definition of Pointwise Minimum of Real-Valued Functions

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \paren{\map f s + \map g x - \size{\map f x - \map g x} }\)

Min Operation Representation on Real Numbers

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \paren{\map f s + \map g s - \size{\map {\paren{f - g} } s} }\)

Definition of Pointwise Difference of Real-Valued Functions

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \paren{\map f s + \map g s - \map {\size{f - g} } s}\)

Definition of Absolute Value of Real-Valued Function

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \paren{\map {\paren{f + g} } s - \map {\size{f - g} } s}\)

Definition of Pointwise Addition of Real-Valued Functions

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \map {\paren{f + g - \size{f - g} } } s\)

Definition of Pointwise Difference of Real-Valued Functions

\(\ds \)

\(=\)

\(\ds \map {\paren{\dfrac 1 2 \paren{f + g - \size{f - g} } } } s\)

Definition of Pointwise Scalar Multiplication of Real-Valued Function

By definition of equality of mappings:

$f \wedge g = \dfrac 1 2 \paren{f + g - \size{f - g}}$

$\blacksquare$