Composition of Mappings/Examples/Arbitrary Finite Sets

Example of Compositions of Mappings

Let:

\(\ds A\)

\(=\)

\(\ds \set {1, 2, 3}\)

\(\ds B\)

\(=\)

\(\ds \set {a, b}\)

\(\ds C\)

\(=\)

\(\ds \set {u, v, w}\)

Let $\theta: A \to B$ and $\phi: B \to C$ be defined in two-row notation as:

\(\ds \theta\)

\(=\)

\(\ds \binom {1 \ 2 \ 3} {a \ b \ a}\)

\(\ds \phi\)

\(=\)

\(\ds \binom {a \ b} {w \ v}\)

Then:

$\phi \circ \theta = \dbinom {1 \ 2 \ 3} {w \ v \ w}$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.4$. Product of mappings: Example $49$
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.4$: Functions: Theorem $\text{A}.4$