Composition of Product Mappings on Natural Numbers

Theorem

Let $a \in \N$ be a natural number.

Let $\mu_a: \N \to \N$ be the mapping defined as:

$\forall x \in \N: \map {\mu_a} x = x a$

Then:

$\mu_{a b} = \mu_b \circ \mu_a$

Proof

\(\ds \mu_{a b}\)

\(=\)

\(\ds x \paren {a b}\)

Definition of $\mu$

\(\ds \)

\(=\)

\(\ds \paren {x a} b\)

\(\ds \)

\(=\)

\(\ds \paren {\map {\mu_a} x} b\)

Definition of $\mu$

\(\ds \)

\(=\)

\(\ds \map {\mu_b} {\map {\mu_a} x}\)

Definition of $\mu$

\(\ds \)

\(=\)

\(\ds \map {\paren {\mu_b \circ \mu_a} } x\)

Definition of Composition of Mappings

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings: Exercise $7$