CSC 165 SLOG #2: More about logic, conjunction and disjuction (Week 3)

This week, we went over the materials in the course faster than usual. It was quite hard for me to understand the material, so I had 1 hour of review time after the lecture sessions. I first learnt about natural language again which I am very familiar with due to my practice on this topic last week.

The professor told us that when we see 'unless' in English, translating it to 'if not' will make our understanding much easier. I also learnt about idiom which was fairly easy.

I also learnt about conjunction and disjunction. One is "and" and the other is "or" These are materials that I already learnt in high school.

For A(x) and B(x) to be true, both of them have to be true. If there is any counter-example to this claim, the claim is false.

For A(x) or B(x) to be true, at least one of them has to be true. If there is no true example, then the claim is false.

I made a chart to demonstrate symbol versions for 'and' and 'or'.

	And	Or
Union Symbol	∩	∪
Logical Symbol	∧	∨

The logical symbol for 'And' is called caret and 'Or' is called reversed caret.

Both union and logical symbols are acceptable as they have the same meaning.

The professor also taught that English can be tricky for conjunction and disjunction. There are inclusive and exclusive. Inclusive means X or Y or both whereas exclusive means X or Y, but not both. This was very interesting to me as there is no inclusive and exclusive situation in Korean. When we want both X and Y, we say X and Y. But if we want X or Y, we say X or Y.

Next thing we learnt is negation symbol which is ¬. The symbol means not. I thought I am fine with the negation until I faced special negation idiom part.

The professor gave us the expression shown below. I will number that expression as 1 to be clear.

#1. ∀x⊆X, P(x) ->Q(x)

Let's call the negation of expression #1 is expression #2

#2. ∃x⊆D, P(x) ∧¬Q(x)

What I first expected for the negation of expression #2 was back to expression #1. However, the professor showed us that the negation of expression #2 is not expression #1, but new expression. Let's call that new expression #3.

#3. ∀x⊆D, ¬(P(x) ∧¬Q(x))

This surprised me quite a lot. However, as I tried solving them, I realized the reason why the negation of expression #2 is not #1, but #3. Then the negation of #3 expression is back to #1. As I learn more about logic, it is getting more confusing and interesting.

Going over truth table part was easy. However, what interested me was how 4 sets would look like in the standard Venn diagram. It would have 2^4=16 regions. However, I wasn't able to draw it properly. Drawing 2 or 3 sets is easy whereas 4 sets is so hard.

As I learn more about logic, my way of thinking and dealing with problem has been improved. I hope to expand my knowledge on logic in CSC 165.