Pointwise Addition on Real-Valued Functions is Commutative
Jump to navigation
Jump to search
Definition
Let $S$ be a set.
Let $f, g: S \to \R$ be real-valued functions.
Let $f + g: S \to \R$ denote the pointwise sum of $f$ and $g$.
Then:
- $f + g = g + f$
That is, pointwise addition of real-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 Real-Valued Functions | ||||||||||
\(\ds \) | \(=\) | \(\ds \map g x + \map f x\) | Real Addition is Commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {g + f} } x\) | Definition of Pointwise Addition of Real-Valued Functions |
$\blacksquare$