1. The example proof of Result 5.24 was super confusing to me. In fact, I still don't think I get it. But I do understand existence proofs, I think. They are basically the opposite of counterexamples. We prove that there's some element for which the statement is true, and thus, the proof is true. So this works only for proofs that state "there is some element" or something like that. I like these proofs, I think. It's nice to know that there's some proofs that you can prove simply by looking at them and seeing that some value for it would work.
2. Unique. What does it mean to be unique? My parents always told me that I was unique, because there is nobody who is exactly like me in the world. So what does it mean to the math 290 world? It means that there is only ONE value in a certain set that has a certain property. So, the examples in the book illustrate that this doesn't mean you are only working with one variable, it means you're working with one value. (This is probably going to confuse me. Wish me luck.) So you can use multiple variables, you just then have to prove that they are equal to each other, thus really making them the same value.
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