...Two of the expressions make sense...
Solution.
So the trick with the map composition is to evaluate them from "right to left". The expression for (a) of \(k \circ h \circ g \circ f\) could be translated into a sequence of steps as follows:
- Apply map \(A \overset{f}{\longrightarrow} B\)
- Apply map \(B \overset{g}{\longrightarrow} A\)
- Apply map \(A \overset{h}{\longrightarrow} C\)
- Apply map \(C \overset{k}{\longrightarrow} B\)
Notice how the codomain of each step matches the domain of the following step. Our composite map follows a path from \(A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} A \overset{h}{\longrightarrow} C \overset{k}{\longrightarrow} B\text{,}\) so it’s domain is the "source" \(A\) and codomain is the "destination" \(B\text{.}\) Alternatively written, \(A \xrightarrow{k \circ h \circ g \circ f} B\text{.}\)
Moving on to (b), something interesting happens when we attempt to list out the steps of \(k \circ f \circ g\text{:}\)
- Apply map \(B \overset{g}{\longrightarrow} A\)
- Apply map \(A \overset{f}{\longrightarrow} B\)
- Apply map \(C \overset{k}{\longrightarrow} B\)
Notice how the codomain for step 2 of \(B\) is not the same as the domain of \(C\) in step 3. This map composition is not well defined because the map \(k\) will not have a valid input to work with.
Having already identified the one that "doesn’t make sense", let’s move directly onto identifying the domain and codomain of the final composition (c) \(g \circ f \circ g \circ k \circ h\text{.}\) Applied from right to left, this gives us:
\begin{equation*}
A \overset{h}{\longrightarrow} C \overset{k}{\longrightarrow} B \overset{g}{\longrightarrow} A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} A
\end{equation*}
As expected, the domains and codomains link up correctly at each step. This means that the domain of \(g \circ f \circ g \circ k \circ h\) is \(A\) (same as the domain of the first map applied, \(h\)) and the codomain is \(A\) (same as the codomain of the last map applied, \(g\)).
