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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
Bautista () Mathematical Investigations April 17, 2008 2 / 45
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
Bautista () Mathematical Investigations April 17, 2008 3 / 45
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?
Bautista () Mathematical Investigations April 17, 2008 4 / 45
Methods of Proof Direct Proof
Direct ProofExample
Prove that for every positive integer n, we can find n consecutivecomposite integers.
Bautista () Mathematical Investigations April 17, 2008 5 / 45
Methods of Proof Direct Proof
Direct ProofExample
If a, b and c are distinct rational numbers, prove that
1
(a− b)2+
1
(b − c)2+
1
(c − a)2
is always the square of a rational number.
Bautista () Mathematical Investigations April 17, 2008 6 / 45
Methods of Proof Direct Proof
Direct ProofExample
Prove that there is one and only one natural number n such that
28 + 211 + 2n
is a perfect square.
Bautista () Mathematical Investigations April 17, 2008 7 / 45
Methods of Proof Direct Proof
Direct ProofSome Combinatorial Examples
(n
r
)=
(n
n − r
)
Bautista () Mathematical Investigations April 17, 2008 8 / 45
Methods of Proof Direct Proof
Direct ProofSome Combinatorial Examples
(n
r
)=
(n − 1
r − 1
)+
(n − 1
r
)
Bautista () Mathematical Investigations April 17, 2008 9 / 45
Methods of Proof Direct Proof
Direct ProofSome Combinatorial Examples
(m
0
)(n
r
)+
(m
1
)(n
r − 1
)+ · · ·+
(m
r
)(n
0
)=
(m + n
r
)
Bautista () Mathematical Investigations April 17, 2008 10 / 45
Methods of Proof Proof by Contradiction
Proof by ContradictionExample
Prove that the number of primes is infinite.
Bautista () Mathematical Investigations April 17, 2008 11 / 45
Methods of Proof Proof by Contradiction
Proof by ContradictionExample
Prove that√
2 is irrational.
Bautista () Mathematical Investigations April 17, 2008 12 / 45
Methods of Proof Proof by Contradiction
Proof by ContradictionExample
Prove that there are no integers x > 1, y > 1 and z > 1 with
x! + y ! = z!.
Bautista () Mathematical Investigations April 17, 2008 13 / 45
Methods of Proof Proof by Contradiction
Proof by ContradictionExample
Given that a, b, c are odd integers, prove that the equation
ax2 + bx + c = 0
cannot have a rational root.
Bautista () Mathematical Investigations April 17, 2008 14 / 45
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.
Bautista () Mathematical Investigations April 17, 2008 15 / 45
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.
Bautista () Mathematical Investigations April 17, 2008 15 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
1 + 2 + · · ·+ n =n(n + 1)
2
Bautista () Mathematical Investigations April 17, 2008 16 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
Show that 5n + 6 · 7n + 1 is divisible by 8.
Bautista () Mathematical Investigations April 17, 2008 17 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
Prove the binomial theorem:
(a + b)n =∑
i
= 0n
(n
i
)an−ibi .
Bautista () Mathematical Investigations April 17, 2008 18 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
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:
Bautista () Mathematical Investigations April 17, 2008 19 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
Scratchwork:For a 2× 2 square:
Bautista () Mathematical Investigations April 17, 2008 20 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
Scratchwork:For a 4× 4 square:
Bautista () Mathematical Investigations April 17, 2008 21 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
Scratchwork:A 2n+1 × 2n+1 square may be divided into four 2n × 2n squares as follows:
2n
2n
2n
2n 2
n
2n
2n
2n
Bautista () Mathematical Investigations April 17, 2008 22 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
Scratchwork:A 2n+1 × 2n+1 square may be divided into four 2n × 2n squares as follows:
2n
2n
2n
2n 2
n
2n
2n
2n
Bautista () Mathematical Investigations April 17, 2008 23 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
Scratchwork:A 2n+1 × 2n+1 square may be divided into four 2n × 2n squares as follows:
2n
2n
2n
2n 2
n
2n
2n
2n
Bautista () Mathematical Investigations April 17, 2008 24 / 45
Methods of Proof Mathematical Induction
The Principle of Mathematical InductionExample
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:
Bautista () Mathematical Investigations April 17, 2008 25 / 45
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.
Bautista () Mathematical Investigations April 17, 2008 26 / 45
Methods of Proof The Pigeonhole Principle
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.
Bautista () Mathematical Investigations April 17, 2008 27 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
Bautista () Mathematical Investigations April 17, 2008 28 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
Bautista () Mathematical Investigations April 17, 2008 29 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
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?
Bautista () Mathematical Investigations April 17, 2008 30 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
The midpoint of (a, b) and (c , d) is(a + c
2,b + d
2
).
Bautista () Mathematical Investigations April 17, 2008 31 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
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.
Bautista () Mathematical Investigations April 17, 2008 32 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
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.
Bautista () Mathematical Investigations April 17, 2008 33 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
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
1 unit
Bautista () Mathematical Investigations April 17, 2008 34 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
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.
Bautista () Mathematical Investigations April 17, 2008 35 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
Bautista () Mathematical Investigations April 17, 2008 36 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
A B
C
DE
F
Bautista () Mathematical Investigations April 17, 2008 37 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
A B
C
DE
F
Bautista () Mathematical Investigations April 17, 2008 38 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
A B
C
DE
F
Bautista () Mathematical Investigations April 17, 2008 39 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
A B
C
DE
F
Bautista () Mathematical Investigations April 17, 2008 40 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
A B
C
DE
F
Bautista () Mathematical Investigations April 17, 2008 41 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
A B
C
DE
F
Bautista () Mathematical Investigations April 17, 2008 42 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
A B
C
DE
F
Bautista () Mathematical Investigations April 17, 2008 43 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
A B
C
DE
F
Bautista () Mathematical Investigations April 17, 2008 44 / 45
Methods of Proof The Pigeonhole Principle
The Pigeonhole PrincipleExample
A B
C
DE
F
Bautista () Mathematical Investigations April 17, 2008 45 / 45
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