Lecture 1

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V63.0120: Discrete Mathematics Spring 2011

• Will use Blackboard for grades, email listserv for everything else.
  • Feel free to use email listserv to post questions to the group. I'll be reading along and helping out where needed.

Text

Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games

Assessment

Survey

• Math majors, CS majors, Other majors
• Taken algorithms, taken data structures, calculus?
• Taking to satisfy math requirement?
• Familiarity with mathematical reasoning?

Intro

Example [Reasoning]

Note: First part always a fact, then draw conclusions based on that.

• If a triangle has three equal sides, then it has three equal angles. This triangle has three equal sides, therefore it has three equal angles.

• If n is even, then n^2 is even. I know n^2 is even. Therefore n is even.

• If you eat your supper, we will go play miniature golf. You didn't eat your supper, so we won't go play miniature golf.
• If you eat your supper, we will go play miniature golf. We will not go play miniature golf, so you did not eat your supper.

Example [Santa Claus House]

• Draw house, wrongly. (starting add even-degree vertex)
• Draw house, correctly. (starting add even-degree vertex)
• What's the difference? Where do I have to start? Why?
  • Hint at ingoing and outgoing lines
  • How many odd-linecount points can we have?
• Draw two houses, connect at SW,SE,N
  • Can this be drawn in one go?

Example [Monty Hall]

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? —Whitaker/vos Savant 1990

Can we figure out a solution?

Can you identify hidden assumptions?

Assumptions:

• the car is initially equally likely to be behind each door
• the host must open a door showing a goat
• must randomly choose which door to open if both hide goats
• and must make the offer to switch.

"Monty is saying in effect: you can keep your one door or you can have the other two doors." The player therefore has the choice of either sticking with the original choice of door, or choosing the sum of the contents of the two other doors, as the 2/3 chance of hiding the car has not been changed by the opening of one of these doors.

Common goal: Eliminate the possibility for disagreement. We would like to discover a method that lets you discover truth 'mechanically', uncovering hidden assumptions and wrong deductions. Build this method in such a way that it makes sense in the context of the 'real world'.

Describing the Class

• "normally" a CS prerequisite
• here: intro to proof
  • no coverage of algorithms, time complexity
• Focus on exactness and methodical correctness
  • A very mechanical business
• What does the 'discrete' stand for? Opposite?

Overview

Our major goal will be to familiarize ourselves with some of the important tools of discrete mathematics.

• Mathematical language, logic, writing, and proof
• Set theory
• Functions and Relations
• Combinatorics and discrete probability
• Graph theory and trees

Section 1.2: Number Puzzles and Sequences

Ways to write sequences

• 5,7,9,11,13,... (odd numbers)
• 1,9,17,25,33,41, ... (+8)
• 1,4,9,16,25,36, ... (squares)
• 2,6,18,54,... (*3)
• 1,5,14,30,55,91, ... (sum of squares)
• 1,2,3,5,7,11, ... (primes)

First of all: There is not enough information! (remember:exactness) Could always imply a rule switch after n elements.

A sequence is a function that maps the natural numbers (1,2,3,...) into another set, often the real numbers.

Common naming: a_n (like function a(n) in calculus)

• a is the name of the sequence

• n is the index or ordinal number

Ways to write down sequences:

• a_1=1, a_2=3 a_3=5, a_4=7, ... (never done!)

• Can you think of a better way? (recursive)
  • Observe: Need substitution! (Like calc functions)
  • If recursive, what else do we need?
    • Index start?
    • Validity range for
• Can you think of another way? (closed-form)
  • What else do we need?

Can we discover each type of description for the examples above?

Definition A recursive formula for a sequence is a formula where each term is described in relation to (its position and) a previous term (or terms!) of the sequence. This type of description must include enough information on how the list begins for the recursive relationship to determine every subsequent term in the list. This is also sometimes called a recurrence relation.

Definition A closed formula for a sequence is a formula where each term is described only in relation to its position in the list.

Important examples

Example [Fibonacci] Given: F_1=1, F_2=2, F_n=F_{n-1}+F_{n-2}

But why? Even if we accept this result as having fallen from the sky, how could we convince ourselves that it is correct? (Substitute into recursive def.)

Example [Gauss]

Natural-language description?

Trick: increasing/decreasing order

Alt. Trick: triangular geometric view

Remark Which is difficult? Recursive \rightarrow closed? Close \rightarrow recursive?

More notation

Sequences of the form

occur quite often, with different increments. Here is a different way of writing them:

Definition A sequence of the form

may be equivalently written as

or, less formally, as

This is often called sum notation or sigma notation. Does this constitute closed form? The base index (1 above) may vary. Note that in \sum_{i=m}^n a_i, always m\le n, otherwise the sum evaluates to zero.