Week 4: Wrapping up Logic Notation and Starting Proofs

In week4's lecture, I found it difficult to prove or disprove statements written with mixed quantifiers. Sometimes I just cannot come up with any counter-examples. However, through doing the assignment 1, I have figure out some rule to justify the statements, that is: always trying to give a derivation to show equivalence and an interpretation to show non-equivalence. FYI, most of the assignment were asking basic questions, but I kept making little mistakes, like the order of mixed quantifiers and accuracy of explanations.

Week4 learned:

Bi-Implication:
(P Q) equals to "(NOT P) AND (NOT Q) OR Q AND P”
 P AND (NOT Q) OR Q AND (NOT P) is its negation.

note: using distributive law and identity law

Transitivity:
(P <=> Q)=> (P => R); its negation is  (P <=> Q)  AND (NOT (P => R))

Structure of Proof:
Assume.....
    Then...
    Then...
Then...

  Note: to find direct proof of universally quantified “P=>Q", we need to find a chain where "P => R1 => R2 => ... => Rn => Q", with n+1 steps in total

Something interesting:
the real-life proof in class--- "unexpected hanging paradox"

paper folding exercise--- we can predicate the up's and down's by a "pagoda" pattern. Also, me and my partner found out another way: when folding (n+1) time, the up's and down's of the paper is 2*( up's and down's when folding n times) + one up

For example:
n=1 up's and down's: one down
n=2 up's and down's : two downs then one up
n=3 up's and down's : two downs one up, tow downs one up, one up


So worried about today's term test.
The End

Week 3: More Elements of Language of Math

From the third week, I have started practicing some past exams. Since most of these are midterms, i have to leave some questions blank. During the practicing, I found my biggest problem is to manage the correct order of quantifiers (turns out that the order matters!) imprecise orders can lead to completely different statements. I need to practice more to get myself familiar to the language of math.

Week3 learned:


➔ Conjunction: AND, ∧, ∩

➔ Disjunction: OR, ∨, ∪

Negation 

note: the negation sign should apply to the smallest possible part of the expression.

      use parentheses properly to avoid ambiguity

Truth Table

lieu of Venn Diagram when predicates are too many (more than 2)

#of rows = 2^(#of predicates)

can be used to:

-evaluating expressions: T or F

-determining satisfiability: 

P ∧ Q : satisfiable   P ∧ ¬P:contradiction  P ∨ ¬P:universal truth(tautology)

-proving equivalent: we get "P => Q" is equivalent to "(NOT P)  ∨  Q"

Some important laws

-Commutative Law

     P  ∧ Q <=>   Q  ∧ P

     P  ∨ Q  <=>  Q  ∨ P

-Associative Law

      P  ∧ (Q  ∧ R)   <=>  (P  ∧ Q) ∧R

      P  ∨ (Q  ∨ R) <=>  (P  ∨ Q)  ∨ R

-Distributive Law

     P  ∧ (Q  ∨ R)  <=>  (P  ∧ Q) ∨  (P  ∧ R)

     P   ∨ (Q  ∧ R)  <=>   (P  ∨ Q)  ∧ (P  ∨ R)

- De Morgan's Law

 NOT(P ∨ Q）<=>  NOT P ∧  NOT Q  vice versa 

- Identity Law 

P  ∧ (Q  ∨ NOT Q)  <=>  P <=>  P  ∨ (Q  ∧ NOT Q)

- Indempotent Law

P  ∧ P <=>  P <=>  P  ∨ P

Week 2: Learning the Language of Math: More about Quantifiers, Statements etc.

Welcome to my spot! After the first quiz I feel a little stressed, since the logic is harder than it looks.

For visualization with Venn Diagram, sometimes it's sufficient to use only one of  ◯ and X; however, in some case excessive usage of ◯ and X can cause wrong answers. To avoid this kind of mistake, I have to be every careful reading through the questions.

For example, to solve this question on quiz,we want to use the diagram to disprove"all students enrolled in a programming course (P)at UofT are also enrolled in a linguistics course (L)at UofT".

What we need is to show "there is some student enrolled in a programming course is not enrolled in a linguistics course".

One little o on the left side of P is sufficient.

In this new lecture, the prof first taught us two new ways to express quantification: using set relations (⊆, ⊈, ∩, ∅, etc.)and using quantifying functions (q1, q2, etc.)Then, we learned the differences between sentences and statements. What interested me most was implication. Many of every language can  translate to P => Q, such as "Can’t have P without Q", "P requires Q", "For P to be true, Q must / need to be true", "Not P if not Q."...all of those are saying P is a subset of Q. Suddenly I found some of human language is so redundant. 0.0！

Note:=> does not mean causes
     in python functions, return not all means some does not, return not any means NO ONE DOES


Some other things learned:
- for implication, there  are converse,contrapositive...

- A Vacuous Truth is a statements that has neither examples nor counterexamples. For example, for any x∈R, x^2 – 2x + 2 = 0 => x > x + 5)

- Equivalence is a statement that satisfies "P => Q" and "P <= Q" simultaneously.That is P IFF Q, P is necessary and sufficient to Q.  Also we can make some weird ones by forming Vacuous Truths in both directions.

- Idioms are some expressions, which are utilized more common than others, for restricting domains.


Although I only read the lecture and attended one tutorial, I find myself have big interest in "how to think logically". Also, the mates in tutorial asked questions frequently, which promoted me to think and know better about the material I had missed.

Week 1: Haven't joined the class but read the lecture online

I was attracted to the course by only reading slides of the first lecture, because of the neat style and interesting way to talk about logic and how to be precise.

Since human language can be ambiguous but intuitive, the careful choose of words (always prefer which with restricted meanings) and quantifiers is needed to precise. I find it rather fun learning to speak precisely like a computer as a human using quantifiers ∀,∃.

I'm taking STA257 currently and have found several techniques are related even reinforced, for example the use of Venn Diagrams and the notion of being precise.So the course content is not tedious for me.

So far I've liked reasoning and logic. Some questions that look difficult are just combinations of two or more simple questions. Drawing a Venn Diagrams is one of my best ways to help me clear the thoughts.

I'm still trying to enroll the course by signing in ROSI constantly to see if someone drops.:)