Week 4: Bi-Implications, Transitivity, Mix Quantifiers and Proofs

The past week, we continued learning the language of math.

First part of lecture, we mainly focused on the manipulating implications into something else, turning them into disjunctions or conjunctions, according to the question. Using distributive and identity properties, we can translate one statement into another form. Along with this, we also studied the negations of the bi-implications.

Next, in transitivity, we learned that if P implies Q, and Q implies R, then P implies R. The Venn diagram consists of P a subset of Q, and Q and subset of R.

For mixed quantifiers, the professor mentioned the differences in meaning (often complete opposites) if 2 of the universal or quantitative statement are ordered incorrectly. For example: for all x in X, there is some y in Y so that P(x,y) will mean something completely different if written: for a y in Y, all x in X, that P(x,y).

This brought us to the end of the language of math- we learned all the logical notations in this course including:
-quantifiers, statements, predicates
-implications, equivalence
-conjunctions, disjunctions, negation
-Venn diagrams, truth tables
-manipulating laws

With these topics covered, we can comfortably express many things using only symbols instead of words.

The next big topic we covered is Proofs. In order to prove something, we must find a proof and write it up. We need to structure our proof accordingly as well. For a direct proof of a universally quantified implication, we need to find a "chain" from the first expression to its implication. The structure for so is:

Proof for for all x in X, P implies Q:
Assume x in X
    Assume P(x)
       Then R1(x)
       Then R2(x)
        ...
        Then Q(x)
    Then P implies Q
Then for all x in X, P implies Q