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