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18.3.1. Unique Factorization
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Let's start by discussing subsequences. A subsequence is a sequence derived from another sequence where some elements may be omitted but the order of the remaining elements is preserved. Can anyone give me an example of a sequence and its subsequence?
How about the sequence [1, 2, 3, 4]? A subsequence could be [1, 3, 4].
Great example, Student_1! So, notice that subsequences are quite flexible. Now, can someone explain why it’s important to understand whether a subsequence can be increasing or decreasing?
I think it helps identify patterns in numbers. If we know a subsequence is strictly increasing, we can predict its behavior.
Exactly! Patterns in sequences allow us to draw significant conclusions. Now, let’s summarize the key part: any sequence of n + 1 distinct real numbers will contain an increasing or decreasing subsequence of length n + 1.
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Now, let’s define strictly increasing and decreasing sequences clearly. Who can describe a strictly increasing sequence?
That's when each term is less than the term that follows it, like [1, 2, 3, 4].
Correct! And what about strictly decreasing sequences?
Each term is greater than the one that follows. Like [4, 3, 2, 1].
Good job! Now, let's take our understanding further. Given how many distinct values are available, why do we always get a subsequence?
Because if we have more terms than possible maximum lengths of subsequences, it has to fit into both categories somehow!
Right! This is where the pigeonhole principle plays a key role in our proof.
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To prove the claim, we'll use proof by contradiction. If we assume there’s no increasing or decreasing subsequence of length n + 1, what would follow?
We’d assume all subsequences are of length at most n.
Exactly! So, what happens when we map these to pairs of subsequence lengths?
We see there are more values than possible lengths, leading to a contradiction by pigeonhole principle.
Well articulated! We conclude that there must exist a value where the maximum lengths of increasing or decreasing subsequences exceed n. This confirms the existence of our subsequence!
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Now that we understand the theorem, let’s explore a practical example. If we take any five distinct real numbers, can you find a subsequence that’s either increasing or decreasing?
Let’s use the numbers [1, 5, 3, 9, 2]. One increasing subsequence is [1, 3, 9].
Excellent! And what about decreasing?
Maybe [5, 3, 2].
Absolutely! This shows how we can apply our understanding of subsequences in real situations. Summarizing, all sequences of n + 1 distinct values guarantee such subsequences.
Overview
Short Summary
This section explores the concept of subsequences in distinct real numbers, proving that any sequence of distinct real numbers contains a subsequence that is either strictly increasing or strictly decreasing.
Medium Summary
The discussion focuses on establishing that within any arbitrary sequence of n + 1 distinct real numbers, there exists a subsequence of length n + 1 which is either strictly increasing or strictly decreasing. This is achieved through the use of the pigeonhole principle and a proof by contradiction to illustrate the validity of this claim.
Detailed Summary
Unique Factorization
In this section, we delve into an important theorem regarding distinct real numbers. The main claim states that given any sequence of n + 1 distinct real numbers, there is always a subsequence of length n + 1 that is either strictly increasing or strictly decreasing.
Key Concepts
- Strictly Increasing Sequence: Defined as a sequence of terms where each subsequent term is greater than its predecessor, denoted as
a_1 < a_2 < ... < a_k. - Strictly Decreasing Sequence: Defined as a sequence of terms where each subsequent term is less than its predecessor, denoted as
a_1 > a_2 > ... > a_k. - Subsequence: A subsequence allows for selection of terms from a sequence without requiring them to be contiguous. For example, from the sequence
[1, 3, 0, -5, 2, 8], we can form subsequences like[1, 2, 8]or[3, 0].
The proof of the main claim employs the pigeonhole principle alongside reasoning about subsequences of maximum lengths (both increasing and decreasing) to arrive at a contradiction if we assume that all such maximum subsequences are of limited length. This leads to a conclusion that at least one subsequence must meet the stated conditions, substantiating the theorem's validity.
Reference YouTube Videos
Audio Book
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Create a free accountFirst of all what is a strictly increasing sequence? A sequence of the form (a_1, a_2, ...) where a_1 < a_2 < a_3 < ... . Whereas if I have a sequence of the form (a_1, a_2, a_3, ...) where a_1 > a_2 > a_3 ... then it is a strictly decreasing sequence. What does a subsequence mean? A subsequence means here that the values may not be consecutive.
Detailed Explanation
A strictly increasing sequence is defined by each element being greater than the one before it. For example, (1, 2, 3) is a strictly increasing sequence. On the other hand, a strictly decreasing sequence is where each element is less than the preceding element, such as (3, 2, 1). A subsequence is a sequence derived by deleting some elements from another sequence without changing the order of the remaining elements. For instance, from the sequence (1, 3, 0, -5, 2, 8), we can pick (1, 2, 8) as a subsequence by skipping some elements.
Examples & Analogies
Think of picking numbers from a lottery ticket. You have to select certain numbers to form a lucky combination. The original ticket has all the numbers, but you can choose any combination of them in the order they appear, which is similar to forming a subsequence.
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Create a free accountSo what this question basically says is that irrespective of the way your n + 1 distinct real numbers are chosen you always have a subsequence. By that I mean that you have a set of n + 1 values going from left to right but need not be in consecutive locations.
Detailed Explanation
The claim states that within any sequence of n + 1 distinct real numbers, there will always be a subsequence of length n + 1 that is either strictly increasing or strictly decreasing. This means when you arrange these numbers, you can always find a way to select some of them that follow this increasing or decreasing pattern, no matter how you choose the numbers.
Examples & Analogies
Imagine a group of people standing in a line with varying heights. Regardless of how you arrange them, you can always find a smaller group of people who are either consistently taller (increasing) or consistently shorter (decreasing).
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Create a free accountSo let the arbitrary sequence of n + 1 distinct real numbers be denoted by a_1 to a_(n+1). I want to prove this statement for every sequence. To do that, I will use the pigeonhole principle along with proof by contradiction.
Detailed Explanation
The idea behind using the pigeonhole principle is that if you have more 'pigeons' (n + 1 numbers) than 'holes' (n possible lengths of subsequences, both increasing and decreasing), there must be at least one hole that contains at least two pigeons, indicating a repeat in lengths. This leads to a contradiction because each length should be unique and hence, at least one of the subsequences must have a length greater than n, fulfilling the condition.
Examples & Analogies
Consider a scenario where you have 10 different fruits, but only 9 baskets to put them in. Since there are more fruits than baskets, at least one basket must contain at least two fruits, illustrating that when choices exceed distinct values, something must repeat.
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Create a free accountWhat does that mean? That means if I try to pair all l_i and d_i pairs then they can take the values in the range (1,1) to (n,n) namely n possible pairs.
Detailed Explanation
By assuming that each subsequence length is at most n (the maximum distinct numbers), we are implying all lengths of subsequences must fall within this limited range. However, this assumption leads to having more pairs of subsequences than possible distinct lengths, forcing us to reconsider the lengths and inevitably creating a contradiction. This contradiction indicates that our assumption that all lengths are limited was wrong.
Examples & Analogies
Imagine trying to distribute prizes to a limited number of participants in a contest, where the number of prizes exceeds competitors. You’d have to end up with duplicates in prize distribution — thus proving that winners cannot all have unique prizes.
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Create a free accountSo assuming that the statement is false leads to a contradiction, thereby proving that for any sequence of n + 1 distinct real numbers, there will always be a subsequence of length n + 1 that is either strictly increasing or strictly decreasing.
Detailed Explanation
By arriving at this contradiction through the steps outlined, we confirm that the claim is fundamentally true. There is no way to avoid creating a longer subsequence when you have more numbers than the maximum allowed distinct subsequence lengths.
Examples & Analogies
Likening this to finding paths in a city: if a city has more roads (distinct real numbers) than possible routes to reach a destination (subsequences), no matter which paths you take, you will eventually have to revisit or combine paths to cover the distance, reflecting the claim's validity.
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Key concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
- Strictly Increasing Sequence:
Defined as a sequence of terms where each subsequent term is greater than its predecessor, denoted as
a_1 < a_2 < ... < a_k.
- Strictly Decreasing Sequence:
Defined as a sequence of terms where each subsequent term is less than its predecessor, denoted as
a_1 > a_2 > ... > a_k.
- Subsequence:
A subsequence allows for selection of terms from a sequence without requiring them to be contiguous. For example, from the sequence
[1, 3, 0, -5, 2, 8], we can form subsequences like[1, 2, 8]or[3, 0].
The proof of the main claim employs the pigeonhole principle alongside reasoning about subsequences of maximum lengths (both increasing and decreasing) to arrive at a contradiction if we assume that all such maximum subsequences are of limited length. This leads to a conclusion that at least one subsequence must meet the stated conditions, substantiating the theorem's validity.
Examples
Memory aids
When numbers are distinct and you choose a few, a subsequence, it's true, will come just in view.
Imagine a wise owl who sorts colorful marbles. It finds that no matter how many marbles it has, each time, it can combine them to have a rainbow path, always increasing or decreasing, too!
Infinity of distinct numbers means pairs of subsequences appearing: Increase or Decrease, always securing! (I-D-P!)
Flash Cards
Glossary
Subsequence
A sequence derived from another sequence where some elements may be omitted but the order is maintained.
Strictly Increasing Sequence
A sequence where each term is less than the term that follows.
Strictly Decreasing Sequence
A sequence where each term is greater than the term that follows.
Pigeonhole Principle
A principle stating that if more items are distributed than containers, at least one container must hold more than one item.