Week 3

Summary of Material
Week three covered conjunction, disjunction, negation, and implication using predicates. Also, a new visual proof called Truth Tables, a neatly organized table of binary elements True and False.

Challenges/Thoughts
The conjunction and disjunction material were very similar to the previous weeks' lectures on union and intersection of sets and therefore was simple to understand. Negation was interesting because it showed that to negate a quantified statement, you could simply replace the negation with the opposite quantifier. (A negation of a universal statement simply requires an existential statement with the negation symbol removed. ) This is very similar to the first week's material where the focus was to simply prove or disprove a statement; where it was taught that a single counter-example would falsify a whole universal statement. It was interesting to see that the material from this week builds upon its previous weeks' and further expands those concepts.

Example
This week's example will be on negation.

Negation of universal quantifier
Ex. All apples in the basket are red.

We can write this as:
A - the universe of all apples, where apples are represented by x
R(x) - where x, the apple, is red

In semi-formal language:
For all x element of A, R(x)

The negation would become:
Not all apples in the basket are red, or
Not (all x element of A, R(x)) - where the Not negation envelopes the whole statement

This although correct, is not preferable because the negation applies to the whole statement. Instead by looking at the meaning, we can reword it to say that there is at least one apple that is not red.

So the final negation is:

There exists an x of element A, Not R(x) - now the Not negation only deals with the R(x) predicate to mean that x does not have the property R (red).

The negation of this statement is similar to falsifying it (This is because of the negation is true, then it must be false.), but it is not necessary to give an example or necessary to be true as it is just another statement.

Weeks 1 & 2

Summary of Material
Starting this week, the course covered the basic transition from casual language to formal, mathematical language. The material covered included Quantification, Predicates, Implications, and Verification/Falsification.

Challenges/Thoughts
The main focus was to give a sense of what formal language resembled and was not too difficult yet. While there were no difficult concepts covered in this week's class, I found vacuous truths to be interesting. I had always thought that to prove a statement to be true, one would have to show that it is not falsifiable and have to show some cases where the statement does appear to be true. Once learning that a statement need only be unfalsifiable, I was at first doubtful that it would be sufficient proof. However, in logic, there is a direct relationship between truth and false, meaning that False and True resemble the relationship between complement sets. Considering False to be the complement of True, if the existence of a false case cannot be proven, then the set of False, or complement of True must be empty; this leads to the conclusion that not True is false.

Example
The example covered for this week's SLOG will cover the bridging between casual language and formal language.

Lets imagine a house with many cats, fat and small (not fat). In the house, there is one play room where most of the cats are kept.

Casual statement: There are some small cats outside the play room.

This sentence states, in formal language, that there is at least one small cat not inside the play room. This can be quantified by an existential quantifier. (I will not be writing formal expression for this week's SLOG.) We will simply represent the universe in the following manner.

Let C be the universe of all the cats in the house.
Let F be the set of all the fat cats in the universe (house).
Let P be the set of all the cats in the play room

In order to prove this statement, only one example of a cat that is both small and not in the room is sufficient, as some in formal language means one or more. That is, a single existence of a cat that is not in the set F and not in the set P.

In order to falsify this statement, we must prove that no cats in the house are small and outside the room. There must be no cats that are both outside the set F and outside the set P. A cat may not be in either set, but cannot be outside both sets.