How to deal with x + y in floor proofs

During my trying to do assignment 2, I stumbled into a nice way of restating stuff like x + y. Actually y can be a variable or an integer. The following is just a sketch that can prove useful; it assumes y is a variable:

To make it easier, [x] means floor of x

x + y = x  + [x] -[x] + y + [y] - [y]

= x-{x}+ y - {y] + [x] + [y] 

but x - [x] < 1. the same for y (from class notes)

Therefore: x + y < [x] +[y] + 2

How is this useful?

Thy this on the limit proof on the assignment to set d as a function of epsilon.

Post midterm

One of the things that I find frustrating is the typing of the symbols. Latex has too much overhead for a slow typist. Word can be adapted to get all the symbols but it usually takes a lot of cut and paste. However, I found a new way. One can set up symbol shortcuts using ctrl and alt keys. This has helped me a lot lately.

A post midterm topic are mathematical proofs. I still do not get the limit proof. This is the second time it has been presented but I cannot follow it. The other proof that I find difficult is the prime number one. I need to study both much more closely. Perhaps using external sources.

Week after midterm and assigment

As the class thins and tutorial quizzes are not picked up, I wonder whether they know something I should know.

The class is interesting but sometimes a bit confusing. It was interesting how limits could be described symbolically illustrates the difference the position of the quantifiers make. However, it was just one example which took the entire class. I think I would have been better off with short examples such as the one that was asked on the midterm. More than one example cements your understanding but there was no time for a second one.

Now we will go into direct proofs. Not sure what to do there yet.

One of the most difficult things I find in the course are the Venn diagrams. Those dealing with universals are much easier than those dealing with existentials. The problem with existentials is that you cannot pin them down anywhere you can just say that the only sure thing you know is that there are some in a certain area. There are fleeting entities. You can be sure some are in an area but what about elsewhere.

Universals are much easier. You can pin them down to only certain areas. In this way you can definitely say whether expressions are valid or invalid. Just point where one is and the other is not. Even with expressions difficult to pin down one cam always De Morgan them or apply some other transformation. p->Q is an example. It is much easier to treat this as ~PvQ on a Venn diagram.

The way to deal with existentials 'seems' to be by showing counter examples to the expressions. This may make sense from a conceptual point of view but when you actually doing it  feels like cheating. One comes up with one's own conditions - one encapsulates the universe by coming up with a set that helps one give an answer to the problem of validity. One is always left with the thought: "Am I being honest about the set I am creating or is it just a set of convenience"