Pointwise Addition on Rational-Valued Functions is Commutative
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Definition
Let $S$ be a set.
Let $f, g: S \to \Q$ be rational-valued functions.
Let $f + g: S \to \Q$ denote the pointwise sum of $f$ and $g$.
Then:
- $f + g = g + f$
That is, pointwise addition of rational-valued functions is commutative.
Proof
\(\ds \forall x \in S: \, \) | \(\ds \map {\paren {f + g} } x\) | \(=\) | \(\ds \map f x + \map g x\) | Definition of Pointwise Addition of Rational-Valued Functions | ||||||||||
\(\ds \) | \(=\) | \(\ds \map g x + \map f x\) | Rational Addition is Commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {g + f} } x\) | Definition of Pointwise Addition of Rational-Valued Functions |
$\blacksquare$