Pointwise Multiplication on Real-Valued Functions is Commutative

From ProofWiki
Jump to navigation Jump to search

Definition

Let $S$ be a non-empty set.

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

Let $f \times g: S \to \R$ denote the pointwise product of $f$ and $g$.


Then:

$f \times g = g \times f$


That is, pointwise multiplication of real-valued functions is commutative.


Proof

\(\, \displaystyle \forall x \in S: \, \) \(\displaystyle \map {\paren {f \times g} } x\) \(=\) \(\displaystyle \map f x \times \map g x\) Definition of Pointwise Multiplication of Real-Valued Functions
\(\displaystyle \) \(=\) \(\displaystyle \map g x \times \map f x\) Real Multiplication is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \map {\paren {g \times f} } x\) Definition of Pointwise Multiplication of Real-Valued Functions

$\blacksquare$