January 21, 2018 - Math Circle 1.0: Absolute Value & Logics

Logic

Here's the summary of our discussions on logic problems in the past two weeks.

Includes solutions to last week's homework and new homework for this week! 

Absolute Value

first, a motivating exercise on the number line:

 

Absolute Value definition

• defined as distance from the origin
• Use $| |$ to indicate absolute value
• if $x>0$ then $|x|=x$
• if $x<0$ then $|x|=-x$
• if $x=0$ then $|x|=0$

Absolute Value examples

First we went through a few questions like find the absolute value of -1, 10, 0, a zillion. Then:

hint: can x = 1? can x = 0? can x = 3? can x = a zillion?

January 7, 2018 - Math Circle 1.0: The number line

The number line

From Gelfand et al 1990 Method of Coordinates:

Introduction to coordinates

The method of coordinates allows us to define where something is
examples
- airplane dropping a package with food in a disaster area
- satellite launch
- cupid --> address
- chess
- lat, lon defines point on earth sfc

Can convert points / lines into numbers

Dimensions: line = 1, plane = 2

The number line

1. origin
2. positive direction
3. unit of measurement

A(5) located on a number line

Defining a point
- ea. point defined by a single number
- the coordinate 5 is 5 units from origin
- notation: A(5) "point A is at coordinate +5"

Correspondence:
- ea. point on line has one and only one number coordinate
- ea. coordinate has one and only one location on the line

Draw A(5) and B(5)
Draw A(2)
    - hint: does not exist

Exercises

draw number line from -5 to 5
place A(5), B(-4), C(1½) on line

- place M(2); which two coordinates are 2 units from M(2)?

Exercise as written in a student's notebook

.

- if B(b) is to the right of A(a), which is greater (a or b)

- Without using a number line, which of the following points is to the right of the other?

•  A(-3), B(-4)
• A(3), B(4)

• A(a), B(-a)

For A(a), B(-a): there are three conditions to consider (a=0, a>0, a<0)

if a=0, A and B are at same point
if a> 0 ...
if a < 0 ...

Homework 

Which of the following is greater:

• M(x) or N(2x)
• A(c) or B(c+2)
• A(x) or B(x²)

 

Math 1.0 Spring 2018 Overview

Spring 2018 Overview

This semester we will focus on coordinate systems, functions, graphs, and logic.

We will begin working with two texts. David will cover coordinate systems and Qiong will cover logic.

Textbooks

1. The Method of Coordinates by Gelfand, Glagoleva and Krillov (1990)  

The introduction begins thus:





read more below:



2. Logical Labyrinths by Raymond Smullyan (2009)

From the preface:

What is symbolic logic? It is also called “mathematical logic,” though the word “mathematics” here has a totally different meaning than the mathematics usually taught in the schools—arithmetic, algebra and geometry. Mathematical logic, instead of dealing specifically with things like numbers, lines and points, deals rather with propositions and arguments, but codified in symbolic forms to facilitate more precise reasoning...

As valuable as symbolic logic is, I have learned from much teaching experience that it is best to begin by devoting some time to informal reasoning. The logic of lying and truth-telling is admirably suited to this, being both entertaining and highly instructive; so this is how we begin in Part I of this book, in which you follow the remarkable journey of the anthropologist Abercrombie through successively more labyrinthine lands of more and more curious liars and truth-tellers of the most varied and sundry sort, culminating in one grand discovery that generalizes all the problems of the preceding adventures. This prepares the readerfor Part II, which begins the formal study of symbolic logic. Part II begins with the subject known as propositional logic, shows how this logic formalizes the reasoning of the earlier chapters, and then advances to the main subject of this book, which is first-order logic, a subject that has many important applications in philosophy, mathematics and computer science...

This book can be used as a textbook for a one- or two-semester coursein logic. A pre-publication copy has been successfully used in this manner at Harvard.