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.