Mathematical Investigations

Mathematical InvestigationsMethods of Proof

Bautista

April 17, 2008

Bautista () Mathematical Investigations April 17, 2008 1 / 45

1 Introduction

2 Methods of ProofDirect ProofProof by ContradictionMathematical InductionThe Pigeonhole Principle

Introduction

The Mathematical Proof

This is the device that makes theoretical mathematics special: the tightlyknit chain of reasoning following logical rules, that leads inexorably to aparticular conclusion. It is proof that is our device for establishing theabsolute and irrevocable truth of statements in our subject. This is thereason that we can depend on mathematics that was done by Euclid 2300years ago as readily as we believe in the mathematics that is done today.No other discipline can make such an assertion. - Krantz, 2007

Methods of Proof

An Example

Into how many regions will n lines, no two of which are parallel and nothree of which are concurrent divide the plane?

Methods of Proof Direct Proof

Direct ProofExample

Prove that for every positive integer n, we can find n consecutivecomposite integers.

Direct ProofExample

If a, b and c are distinct rational numbers, prove that

(a− b)2+

(b − c)2+

(c − a)2

is always the square of a rational number.

Direct ProofExample

Prove that there is one and only one natural number n such that

28 + 211 + 2n

is a perfect square.

Direct ProofSome Combinatorial Examples

n − r

(n − 1

r − 1

(n − 1

r − 1

)+ · · ·+

(m + n

Methods of Proof Proof by Contradiction

Proof by ContradictionExample

Prove that the number of primes is infinite.

Prove that√

2 is irrational.

Prove that there are no integers x > 1, y > 1 and z > 1 with

x! + y ! = z!.

Given that a, b, c are odd integers, prove that the equation

ax2 + bx + c = 0

cannot have a rational root.

Methods of Proof Mathematical Induction

The Principle of Mathematical Induction

Theorem (The Principle of Mathematical Induction)

If a subset M of Z+ (= the set of positive integers) satisfies the conditions

1 1 ∈ M

2 n ∈ M implies that n + 1 ∈ M

then M = Z+.

Proof.

Suppose there is a positive integer not belonging to M. Then, there is asmallest such integer m. But m 6= 1 since [1] states that 1 ∈ M. Thus,m < 1. Now, consider m− 1. If m− 1 ∈ M, then m ∈ M which leads to acontradiction. If m − 1 /∈ M, then we contradict minimality of m. Thus,there can be no such m.

The Principle of Mathematical Induction

Theorem (The Principle of Mathematical Induction)

If a subset M of Z+ (= the set of positive integers) satisfies the conditions

1 1 ∈ M

2 n ∈ M implies that n + 1 ∈ M

then M = Z+.

Proof.

Suppose there is a positive integer not belonging to M. Then, there is asmallest such integer m. But m 6= 1 since [1] states that 1 ∈ M. Thus,m < 1. Now, consider m− 1. If m− 1 ∈ M, then m ∈ M which leads to acontradiction. If m − 1 /∈ M, then we contradict minimality of m. Thus,there can be no such m.

The Principle of Mathematical InductionExample

1 + 2 + · · ·+ n =n(n + 1)

Show that 5n + 6 · 7n + 1 is divisible by 8.

Prove the binomial theorem:

(a + b)n =∑

)an−ibi .

Prove that for any positive integer n, a 2n × 2n square grid with 1 squareremoved can be covered with L-shaped tiles that look like this:

Scratchwork:For a 2× 2 square:

Scratchwork:For a 4× 4 square:

Scratchwork:A 2n+1 × 2n+1 square may be divided into four 2n × 2n squares as follows:

Suppose n is a positive integer. An equilateral triangle is cut into 4n

congruent triangles and one corner is removed. Show that the remainingarea can be covered by red trapezoidal tiles like those shown in the figure:

Methods of Proof The Pigeonhole Principle

The Pigeonhole Principle

If kn + 1 objects (k ≥ 1) are distributed among n boxes, one of the boxeswill contain at least k + 1 objects.

The Pigeonhole PrincipleExample

Consider a 3× 7 rectangle divided into 21 squares as shown below. If allthe squares are to be colored either red or blue, show that no matter howthese squares are colored, one will always form a rectangle whose cornersare all of the same color.

If we are to look at the board by columns, then we only have eightpossible columns as shown below. When will a rectangle of vertices withthe same color be formed?

The midpoint of (a, b) and (c , d) is(a + c

2,b + d

If any five of the infinite points shown above are chosen. Show thatthere will always be two of the five points whose midpoint is a latticepoint.

Suppose A is a set of 19 numbers chosen from the numbers

1, 4, 7, 10, 13, . . . , 97, 100.

Show that no matter how A is selected, there will always be twowhose sum is 104.

If 5 points are put inside asquare of side 1 unit, showthat no matter how thesepoints are located, therewill always be two whosedistance between them isless than or equal to

√2/2.

1 unit

Given 6 points, no three of which are collinear, show that if all the 6points are joined with each other by blue or red segments then no matterhow the segments are colored, a triangle with sides of the same color willalways be formed.

Mathematical Investigations

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