Week 3 Logic

In week 3, we learned logic, which is using symbols to present logical statements. The symbols include conjunction and disjunction etc.

Conjunction, which is "∧", is similar to intersection and is the expression of and in logic. Because conjunction is similar to intersection, conjunction is false if either part is false. For example, for x∈prime number, x = 3 ∧ x = 7 is true, but x = 3 ∧ x = 4 is false.

Disjunction, which is "∨", is similar to union and is the expression of or in logic. In disjuction, it's different from conjunction, disjunction is true if either part is true and is false if both parts are false.

Then, we learned negation, whose symbol is "¬". Negation is the nagated original statement. The original statement is true if its negation is false.

We also learned some standard transformations for negated expressions like  "¬ ∀x ∈ X, .... ⇔ ∃ x ∈ X, ¬....", and then we can use them to negate a statement as well as change the "¬" symbols position in a statement. 

Drawing truth and tabulating truth are two ways to check if an implication is true. We draw Venn diagrams for drawing truth and build tables for all possible outcomes for tabulating truth.

Week2 Quantifiers

In the first two weeks, I learned CSC165 is the course that help us to understand programming and math languages, which needs precision.

Then, the main thing we learned is quantifies.Quantifies are symbols including a variable that indicates the degree of generality of the expression in which the variable occurs. The symbols include Universal Claim(∀), existential claims(∃), negation(¬) etc.

The sentence comprised by the symbols is called statement. And there is also a negation for the statement. For example, if the statement states that "there is no female students in computer science specialist". Then, the negation will be "there exist at least one female student in computer science specialist", but not "there is no male student in computer science specialist." 

The statements also need to be precise. For example, the sentence " The students go to school in the morning." is not precise enough. Because "the students" doesn't show the number of students and the information for students either. "The school" doesn't show which school it is and "morning" doesn't show the exact time.

Then, the converse, contrapositive and negation of a statement is defined. Converse is the reverse of the original implication. Contrapositive is the toggled and reversed implication. Negation is the statement which false the original sentence. 

eg. the original statement is: ∀ x ∈ E, F(x) ⇒ L(x)
      the converse of it is: ∀ x ∈ E, L(x) ⇒ F(x)
      the contrapositive of it is: ∀ x ∈ E, ¬ L(x) ⇒ ¬ F(x)
      the negation of it is: ∃ x ∈ E, F(x) ⇒ ¬ L(x)