Several parts of this section were difficult to understand. The introduction of a restriction went fine, however when applied to the Reals, that threw me. Once I reread it and understood that the restriction is a subset of the initial function, or rather that the restriction is taken on a set that is a subset of the initial domain, I understood how the restriction g1 could be one-to-one. Effectively we are only taking the positives and thus leaving out the negatives that would yield the same values of the range when squared.
Lemma 10.16 is still throwing me and I would appreciate it if we spent some time on it in class. I don't understand why we take values that are in the union of A and B. It seems weird to be taking values from the domain and range of a function.
The definition of a function from the union of two sets to the union of their corresponding sets that make up the range of two individual functions was super cool. I hadn't really thought about that before. It also makes sense that the two sets A and C would need to be disjoint for the function h to be considered a function (pg 237).
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