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