Section 3.8 Session 8: Pictures of a map
Rather than having exercises of its own, this session was focused on solving the exercises from Article II. Since I’d already worked out these problems, all that remains for this week is to proofread.
Example 3.8.1. Article II, Exercise 5 Proofreading.
(Add the rest of the arrows yourself.)Solution.
My diagram of all 6 sections from earlier matchs up with this incomplete diagram perfectly:
Figure 3.2.4Likewise, I correctly arrived at a count of 8 retractions and drew those too:
Figure 3.2.7If there’s something to be critical about myself here, it would be that I was perhaps a little lax in assuming the domains and codomains of my maps were the same. I jumped right into diagrams of \(g\) without specifying what it was. In my defense, this information was implicit in the diagrams themselves but the authors took an extra effort to explicitly articulate the qualities they were checking for in the map.
Example 3.8.2. Article II, Exercise 8 Proofreading.
Prove that the composite of two maps that have sections has a section.Solution.
I must admit, it took me several attempts to verify that my solution was equivalent to the book’s with direct substitutions. I had used \(\set{f_1, s_1, f_2, s_2}\) instead of the respective \(\set{k, s', p, s}\text{.}\)
Figure 3.3.7I think I actually prefer my notation here over the authors because it makes the connection between the functions and sections more clear. I kept forgetting whether \(s\) or \(s'\) matched up with \(p\) or \(k\text{.}\) Using the \(\set{f_1,f_2}\) notation makes the reversed composition ordering of the sections more readily apparent.
Based on what I saw in this session, I think I was on the right track with my solutions to the exercises from Article II. It’s nice to have this little checkpoint to validate my earlier work. I’m usually the type that likes to try to solve the exercises in my head just to see if I can, but then I miss out on moments for reflection like this.
