Proof Structures

Given a statement to prove, the proper way to start off the proof is to assume the variables that are assigned the "for all" quantifier are true.
Then for the following variables (say x) assigned the existential quantifier, you have to find an actual example that belongs in the set that satisfies the conditions.
It is sometimes very difficult to find an example that will fit into the proof, so it may be easier work backwards.
That's why in the proof, we say variable x is equal to, in the beginning, an unknown element.

We start at the implication and rearrange the antecedent in such a way that resembles to what we are trying to prove in the consequent.
Most of the time, x can be easily found from the scratch work done from the process of rearranging the antecedent.

Going back to the proof structure, we now know what x is, and so now we can finish the full proof with an x that works, and that will complete the proof.

Week 5

Just completed week 5. I've done numerous quizzes (which were pretty good) and a term test.

The test was manageable, but I wasn't think through the last question properly. I had to create 3 sets to prove one statement false and another true. I set one of them to be even numbers and the other to be odd. I don't know why I didn't just sign a few numbers to each of the sets. Halfway through the question, I realized I interpreted the statements wrong but I had no time to change it. So that was pretty upsetting.

Apart from that question, the rest of the test was pretty good.

We are starting to go into proving mathematical claims now, which was never really something that I was good at.
What I've gotten from class so far is to assume the things stated in the antecedent, and somehow use that and tweak it so that it resembles the consequent. It might sometimes be easier to try and work out the consequent first so that we know what we should be doing to the antecedent.

Lastly, while studying for the term test, I actually tried to figure things out with truth tables and yes, it works.

After doing 3 tutorial exercises and an assignment, I've found out that it makes it much easier to translate sentences given to me into symbols. If I assign quantifiers, implications, conjunctions and disjunctions, it is much more easier to solve the negation, converse, or contrapositive of the claim.

When I'm supposed to negate a sentence, I know I have to write it so that the sentence becomes false if it was originally true, but it's much easier to see when it's written symbolically.