MATH 109 Lecture 1

Textbook: An Introduction to Mathematical Reasoning - Peter Eccles Textbook ISBN: 0521597188

Rough Overview

Weeks 1-2: Basic Proof Theory (Ch. 1-5)

• Proof by contradiction
• Mathematical induction (important)

Week 3-4: Sets and Functions (Ch. 6-?)

• End of midterm 1 content

Elementary Number Theory

Combinatorics - counting things

• End of midterm 2 content

Chapter 1

Statement

A (mathematical) statement is a sentence that asserts something and is either true or false.

Today is September 30

False statement

$3>2$

True statement

$12+11$

Not a statement

$12+11=23$

True statement

If $a>b$ and $b>0$ then $a^2>b^2$.

Involves “and,” “if,” “then” Involves “variables” ,

A statement is atomic if it cannot be broken up into smaller ones

More complicated mathematical statements are made from combining atomic statements with logical connectors.

Logical “not”, “and”, and “or”

(next class “imply” and “equivalent” (Ch. 2))

not () (negation) and () (conjunction) or () (disjunction)

From now on, , , represent statements

• - negation of
• - and
• - or

Concrete exercise Let be Let be - It is false that - &

Truth Table for Locial Connectors

Truth Table - deducing the truth value of complicated mathematical statements using the truth values of atomic statements and truth rules

Truth Table for $\wedge$

T	T	T
T	F	F
F	T	F
F	F	F
The last column represents the truth values for

Truth Table for $\neg$

T	F
F	T

Truth Table for $\vee$

T	T	T
T	F	T
F	T	T
F	F	F

Suppose $P\vee Q$ is true

Some people might think that exactly one of is true. However, it just means they are not both false.

Summary

What is a mathematical statement? Can build complicated statements using logical connectives. Properties of logical statements are given by truth tables. Truth values of complicated statements can be determined if we know the truth values of the atomic statements.

$(\neg P)\vee Q$

T	T	T
T	F	F
F	T	T
F	F	T