Week 4: Many proves

In this week's lecture, the lecture is mainly about proof.

Although we have all practiced how to prove in high school, this week's lecture is a little bit different. When we get something to prove, the first thing to do is to understand the reason that makes you believe what you are proving is correct. Then is to write down the expression. As we all know, the key point here is to be correct and can convince other people.

I have never learnt about the structure to prove something until this week's lecture. First, write the "for all" or "exist" quantifier sentence (I don't know the term to describe that) on the first few lines. Then in the next line or few lines, write the antecedent. What should we notice here is that, since the assumption is kind of like "contains" in the previous sentence in logic, you should leave some space from the beginning of the line, just like the python language (by pressing Tab). After writing down the assumption, you need to focus on the proving. Maybe you want to prove P(x) => Q(x), but usually you can get there immediately, so prove P(x) => R1(x), then R1(x) => R2(x), ......, finally Rn(x) => Q(x). According to the transitivity, you can conclude that P(x) => Q(x).

There is another topic in this week, which is about whether two quantifiers can be swapped. The professor take the definition of limit as an example which makes me feel a little bit like being in MAT137 class. But the point in this class is not to learn the definition, but to learn that if the universal quantifier and the exist quantifier are exchanged, there will be different meanings, and the definition after the change is incorrect. However, it a proposition contains only universal or only exist quantifier, then they can be exchanged and they express the same meaning.

I found that many of us do not use symbols in slog. For example, here. I believe it is quite easy in word to type "\exists" to get "∃" and type "\in" to get "∈". Using LaTeX is also a good choice. Using words instead of symbols is sometimes confusing, and if you don't want to use symbols, you should state the complete sentence.