Discrete Mathematics

Discrete Mathematics

WEN-CHING LIEN Department of Mathematics National Cheng Kung University

2008

5.5: The Pigeonhole Principle

The Pigeonhole Principle:

If m pigeons occupy n pigeonholes and m >n, then at least one pigeonhole has two or more pigeons roosting in it.

Example (5.39)

An office employs 13 file clerks, so at least two of them must have birthdays during the same month.

Here we have 13 pigeons (the file clerks) and 12 pigeonholes (the months of the year).

5.5: The Pigeonhole Principle

The Pigeonhole Principle:

If m pigeons occupy n pigeonholes and m >n, then at least one pigeonhole has two or more pigeons roosting in it.

Example (5.39)

An office employs 13 file clerks, so at least two of them must have birthdays during the same month.

Here we have 13 pigeons (the file clerks) and 12 pigeonholes (the months of the year).

5.5: The Pigeonhole Principle

The Pigeonhole Principle:

If m pigeons occupy n pigeonholes and m >n, then at least one pigeonhole has two or more pigeons roosting in it.

Example (5.39)

An office employs 13 file clerks, so at least two of them must have birthdays during the same month.

Here we have 13 pigeons (the file clerks) and 12 pigeonholes (the months of the year).

Let S ⊂Z⁺,where|S| =37.

Then S contains two elements that have the same remainder upon division by 36.

Example (5.47)

Let m∈Z⁺,with m odd.

Prove that there exists a positive integer n such that m divides 2ⁿ−1

Example (5.42)

Let S ⊂Z⁺,where|S| =37.

Then S contains two elements that have the same remainder upon division by 36.

Example (5.47)

Let m∈Z⁺,with m odd.

Prove that there exists a positive integer n such that m divides 2ⁿ−1

Let us start by considering two particular examples:

1 Note how the sequence 6,5,8,3,7 (of length 5) contains the decreasing subsequence 6,5,3 (of length 3).

2 Note how the sequence 11,8,7,1,9,6,5,10,3,12 (of length 10) contains the increasing subsequence 8,9,10,12 (of length 4).

These two instances demonstrate the general result:

For each n∈Z⁺, a sequence of n²+1 distinct real numbers contains a decreasing or increasing subsequence of length n+1.

To verify this claim let a₁,a₂, . . . ,a_n2+1be a sequence of n²+1 distinct real numbers.

...continued

Example (5.49)

Let us start by considering two particular examples:

1 Note how the sequence 6,5,8,3,7 (of length 5) contains the decreasing subsequence 6,5,3 (of length 3).

2 Note how the sequence 11,8,7,1,9,6,5,10,3,12 (of length 10) contains the increasing subsequence 8,9,10,12 (of length 4).

These two instances demonstrate the general result:

For each n∈Z⁺, a sequence of n²+1 distinct real numbers contains a decreasing or increasing subsequence of length n+1.

To verify this claim let a₁,a₂, . . . ,a_n2+1be a sequence of n²+1 distinct real numbers.

...continued

Let us start by considering two particular examples:

1 Note how the sequence 6,5,8,3,7 (of length 5) contains the decreasing subsequence 6,5,3 (of length 3).

2 Note how the sequence 11,8,7,1,9,6,5,10,3,12 (of length 10) contains the increasing subsequence 8,9,10,12 (of length 4).

These two instances demonstrate the general result:

For each n∈Z⁺, a sequence of n²+1 distinct real numbers contains a decreasing or increasing subsequence of length n+1.

To verify this claim let a₁,a₂, . . . ,a_n2+1be a sequence of n²+1 distinct real numbers.

...continued

Example (5.49)

Let us start by considering two particular examples:

1 Note how the sequence 6,5,8,3,7 (of length 5) contains the decreasing subsequence 6,5,3 (of length 3).

2 Note how the sequence 11,8,7,1,9,6,5,10,3,12 (of length 10) contains the increasing subsequence 8,9,10,12 (of length 4).

These two instances demonstrate the general result:

For each n∈Z⁺, a sequence of n²+1 distinct real numbers contains a decreasing or increasing subsequence of length n+1.

To verify this claim let a₁,a₂, . . . ,a_n2+1be a sequence of n²+1 distinct real numbers.

...continued

Let us start by considering two particular examples:

1 Note how the sequence 6,5,8,3,7 (of length 5) contains the decreasing subsequence 6,5,3 (of length 3).

2 Note how the sequence 11,8,7,1,9,6,5,10,3,12 (of length 10) contains the increasing subsequence 8,9,10,12 (of length 4).

These two instances demonstrate the general result:

For each n∈Z⁺, a sequence of n²+1 distinct real numbers contains a decreasing or increasing subsequence of length n+1.

To verify this claim let a₁,a₂, . . . ,a_n2+1be a sequence of n²+1 distinct real numbers.

...continued

Example (5.49)

Let us start by considering two particular examples:

1 Note how the sequence 6,5,8,3,7 (of length 5) contains the decreasing subsequence 6,5,3 (of length 3).

2 Note how the sequence 11,8,7,1,9,6,5,10,3,12 (of length 10) contains the increasing subsequence 8,9,10,12 (of length 4).

These two instances demonstrate the general result:

For each n∈Z⁺, a sequence of n²+1 distinct real numbers contains a decreasing or increasing subsequence of length n+1.

To verify this claim let a₁,a₂, . . . ,a_n2+1be a sequence of n²+1 distinct real numbers.

...continued

For 1≤k ≤n +1,let

x_k =the maximum length of a decreasing subsequence that ends with a_k,and

y_k =the maximum length of an increasing subsequence that ends with a_k.

For 1≤k ≤n²+1,let

x_k =the maximum length of a decreasing subsequence that ends with a_k,and

y_k =the maximum length of an increasing subsequence that ends with a_k.

For 1≤k ≤n +1,let

x_k =the maximum length of a decreasing subsequence that ends with a_k,and

y_k =the maximum length of an increasing subsequence that ends with a_k.

Thank you.