Week 2 and 3. What did we learn?

This two weeks, I've learnt that we can mesh together logic and set theory, which seemed to be mutually exclusive at first. For example, take the following sentences.

a implies b. In set theory, this means that a is a subset of b. This becomes apparent when we consider venn diagrams as the set of a by itself must be empty, implying that all of a must exist inside b, thus making 'a' a subset of b.

a and b. In set theory, this means that the set that makes this true is the intersection between a and b. Apparent through venn diagrams and no set theory, I can understand more about logic.

a or b. In set theory, like 'and', the set theory operator is union, which is everything that includes anything within a or b, or maybe both. This even applies for mutually exclusive items!

As I stated in the last post, I found vacuous truths to be one of the more confusing parts about logic. However, using my understanding of when an implication is true, and also using set theory, I am able to verify that a vacuous truth will always make an implication true. To make anything ever false, you must find some existing counterexample. In the case of a vacuous truth, we will never find a counter example! Then, using an explanation through set theory, a vacuous truth implies an empty set. An empty set is always a subset of any other set. That's why a vacuous truth will always be true.

Finally DeMorgan's law presented a new way for statements to change. After doing truth tables, I wondered what if some expression had the same truth values as an if statement, or an and statement? Would they be considered "equivalent"? As De Morgan's law shows, if they have the same truth values, then they actually are the exact same as each other; they are equivalent.

Finally, the experiment on Friday with paper folding actually brought to my attention how interesting it is to approach a problem and solve it just through experimentation. Usually, I'd expect there to be some manipulation you have to pull before being able to solve it. Now, I realize that we have to experiment, play around, and try to understand a bit more before being able to identify that crux move that leads to a solution.