Week 3

Here are what I learn in week 3...

Conjunction (AND, ∧) noun
“the action or an instance of two or more
events or things occurring at the same point
in time or space.”
To put this in perspective, P(x) ∧ Q(x) means
1)P(x) intercept Q(x)
2)P(x) = True and Q(x) = True
3)The interlocked part of the circles on a Venn Diagram of P(x) and Q(x)

Disjunction (OR, ∨)
"Combine two statements by claiming that at
least one of them is true."
To put this in perspective
1)P(x) union Q(x)
2)P(x) = True, Q(x) = True, or P(x),Q(x) both equal to True
3)Both circles of the Venn Diagram of P(x) and Q(x)
If you want to mean P(x) or Q(x) but not both - that's exclusive-or!

Negation (NOT, ¬)
NOTHING fancy about it, get it?
Literally mean NOT
BUT, you can move a negation further into a bracket by changed the universal quantifier into existential quantifier and vice versa
example:
1)NOT(all Koreans are good at playing League of Legends) = there exist a Korean who is NOT good at playing league of Legends
2)NOT(there exist a chicken who loves eating dog) = all chicken NOT love eating dog.

Parenthesis
1) they are important - you basically need them for every connectives you have (conjunction, disjunction, implication, etc)
2) what happens in the parenthesis stays in the parenthesis (you are have two xs in two parenthesis and they won't mean the same thing)

Truth Table
"Enumerate the outputs over all possible
combinations of input values of P, Q" - and how many other variables you have.
In this case
Input = either True or False
P, Q... are all predicates
You draw a table, where your predicates are in the first few columns, and the predicate logic you want to check are in the following columns.
Now base on the combination of True and False of your variables, you check whether the predicate logic involving those variable are True or False.
It has as many rows are 2 to the power of the number of predicates you have.
1) If a column is all True, then the predicate logic is a tautology
2) If a column is all False, then the predicate logic is unsatisfiable
3) Otherwise, it is satisfiable
4) If two column have the same truth values, then they are equivalent

De Morgan's law
Basically, if you have a negation of a conjunction, NOT(P AND Q)
                                      it can be turn into NOT(P) OR NOT(Q)
similarly NOT(P OR Q) is equivalent to NOT(P) AND NOT(Q)
*Note, you just break the bracket, apply negation to both elements, and change AND to OR

Laws 
*For a full list of laws to be used in CSC165,
read Chapter 2.17 of Course Notes
But for those who want to know some
Conjunction and disjunction have the following laws
1) commutative - P AND/OR Q is equivalent to Q AND/OR P
2) associative - where put the bracket is not important for the same sign
3) distributive - P AND (Q OR R) is equivalent to (P AND Q) OR (P AND R)
4) identity - P AND NOT(P) is False, P OR NOT(P) is True
5) idempotent - P OR P, P AND P are equivalent to P

This is all from week 3.............. a little shorter than last week?