Combining Solutions
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Increasing and Decreasing Subsequences
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Today, we are discussing subsequences in sequences of distinct real numbers. Can anyone tell me what a strictly increasing sequence is?
Isn't it a sequence where each term is less than the next one?
Exactly! For example, in the sequence (1, 2, 3, 4), each number is less than the next. Now, what about a strictly decreasing sequence?
That's when each number is greater than the one that follows, right?
Correct! Like in the sequence (4, 3, 2, 1). Great work! Now, let’s see how we can find such subsequences in a larger context.
Definition and Importance of Subsequences
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
A subsequence means picking elements from the original sequence while skipping some. For example, if we have the sequence (1, 3, 0, -5, 2, 8), can you give me a subsequence?
How about (3, 2, 8)?
Exactly! And why is identifying subsequences important?
Because it helps us understand the overall pattern in a sequence, right?
Precisely! Now, let's discuss how every sequence of k+1 distinct real numbers guarantees a subsequence of length k+1.
Proof by Contradiction and Pigeonhole Principle
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
To prove that a subsequence exists, we can use contradiction. What do we assume?
We assume that the lengths of both increasing and decreasing subsequences are at most k.
Brilliant! And how does the pigeonhole principle fit into this argument?
It shows that with k+1 distinct numbers, we can't have both lengths only being k.
Exactly. It leads us to conclude that at least one subsequence must exceed length k. Great grasp of the concept!
Example Application of the Theorem
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Let's say we have the sequence (3, 7, 5, 1, 6). How would we demonstrate our theorem here?
We can look for increasing and decreasing subsequences among these numbers.
Correct! Can anyone identify at least one strictly increasing subsequence?
How about (3, 5, 6)?
Well done! And now, a decreasing one?
I see (7, 5, 1) as decreasing.
Excellent examples! This confirms our theorem in action!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section explains the existence of a subsequence of length k+1 in any sequence of k+1 distinct real numbers. This subsequence will either be strictly increasing or strictly decreasing, and the proof utilizes principles such as the pigeonhole principle and a contradiction argument regarding pairings of subsequence lengths.
Detailed
Detailed Summary
The claim subjected to demonstration in this section asserts that given any sequence of k+1 distinct real numbers, there exists a subsequence of length k+1 that is either strictly increasing or strictly decreasing. An increasing sequence is defined by a succession where each term is less than the next (e.g., a1 < a2 < ...), while a decreasing sequence entails that each term is greater than the next (e.g., a1 > a2 > ...).
Defining a subsequence, we clarify that terms need not be contiguous, allowing for the selection of non-consecutive elements. The essence of the proof involves identifying the lengths of the longest increasing and decreasing subsequences drawn from the sequence of k+1 distinct real numbers. By utilizing the pigeonhole principle, the existence of a subsequence of the desired length is guaranteed. It involves a contradiction methodology suggesting the possibility of pairing indexed subsequences that share corresponding lengths. Ultimately, through this pairing process, a contradiction arises if both subsequence lengths remain uniformly bounded by k. Thus, at least one subsequence must be longer than k, confirming the stated claim.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Understanding Subsequences
Chapter 1 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Here we want to show the following: that you take any sequence of n + 1 distinct real numbers. They are arbitrary real numbers; may be positive, negative in any order you take them. The only condition is that they have to be distinct. Then the claim is that irrespective of the n + 1 real numbers that you have in your sequence, you always have a subsequence of length n + 1 which is either strictly increasing or strictly decreasing.
Detailed Explanation
In this chunk, we introduce the idea of subsequences formed from a sequence of distinct real numbers. A subsequence can be created from the original sequence by possibly skipping some elements, but maintaining the order of the remaining elements. The claim emphasizes that in every group of n + 1 distinct real numbers, there will always exist either a subsequence that is strictly increasing or one that is strictly decreasing.
Examples & Analogies
Imagine you are organizing a series of books by height. No two books are the same height. If you take any five books (n = 4, because n + 1 = 5), no matter how you arrange them, you can always find a way to select four that are either getting taller or shorter as you go down the line.
Defining Increasing and Decreasing Subsequences
Chapter 2 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
First of all, what is a strictly increasing sequence? A sequence of the form (a_1, a_2, ...) where a_1 < a_2 < a_3 < ... < a_n. Whereas if I have a sequence of the form (a_1, a_2, a_3, ...) where a_1 > a_2 > a_3 ... > a_n then it is a strictly decreasing sequence.
Detailed Explanation
This chunk defines what constitutes strict monotonicity in sequences. A sequence is considered strictly increasing if its terms are in a sequence where each term is less than the next one. Conversely, a sequence is strictly decreasing if each term is greater than the following term. This mathematical distinction is crucial for understanding the later proofs and claims.
Examples & Analogies
Think of a marathon race. If all runners are continuously getting faster, that's like an increasing sequence. If all runners are slowing down, that's like a decreasing sequence. In each case, you can observe that the positioning does not repeat.
The Pigeonhole Principle
Chapter 3 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Let the arbitrary sequence of n + 1 distinct real numbers be denoted by a_1 to a_n+1. To prove a universally quantified statement, we can use the universal generalization principle by proving that the statement is true for some arbitrary element of the domain. My domain here is the set of all possible sequences of n + 1 distinct real numbers.
Detailed Explanation
Here we focus on the Pigeonhole Principle, a fundamental concept in combinatorics. By defining arbitrary elements of our overall problem (sequences of numbers), we can leverage this principle to show that there must be overlap (i.e., at least one subsequence exists that satisfies either increasing or decreasing conditions). It sets the stage for the argument that follows.
Examples & Analogies
Consider a classroom with 30 students and only 5 types of fruit snacks. If every student takes a snack, you will find at least one type of snack that has been chosen by more than one student. This guarantees some overlap, much like finding overlapping subsequences in our sequences of numbers.
Finding Contradictions
Chapter 4 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Assume that for each a_i in the sequence, the value l_i (the length of the longest increasing subsequence starting at a_i) is at most n, corresponding to the maximum length of subsequences we expect from our sequence. Using the Pigeonhole Principle, we deduce contradictions arise when we assume these subsequence lengths do not meet the criteria predicted.
Detailed Explanation
This section discusses how via contradiction, we can assume maximum lengths for increasing and decreasing subsequences. If we exceed the available numbers (pigeons) for these lengths (holes), we must arrive at a conflict, suggesting at least one subsequence must satisfy the increasing or decreasing condition laid out. This establishes the existence of the desired subsequence as a fundamental truth.
Examples & Analogies
Imagine a scenario where you have several boxes (holes) that can each hold a certain maximum number of books (pigeons). If you suddenly find that each box is full, but you have books remaining, at least one box must be overflowing. This realization is akin to finding a subsequence that violates the initial assumption.
Conclusion and Existence of Subsequences
Chapter 5 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
After establishing a contradiction, we can conclude that there must exist at least one a_i such that either the length l (for increasing subsequences) is greater than n or the length d (for decreasing subsequences) is greater than n. This confirms that in any sequence of n + 1 distinct real numbers, there will always be subsequences that meet the specified conditions.
Detailed Explanation
This chunk wraps up the examination by re-enforcing that through the established principles and the argumentation applied, we find either a lengthier increasing or decreasing subsequence must exist, which confirms the claim made at the beginning. It illustrates the broader application of these principles will always yield specific results as stated.
Examples & Analogies
Think of looking at a crowd where everyone wears distinct color shirts. No matter how you sample individuals from this crowd, you will eventually find at least one group that either hums a rising tune (like increasing) or a falling tune (like decreasing), validating the original claim.
Key Concepts
-
Subsequences: A sequence formed by selecting terms from another sequence while maintaining their order.
-
Strictly Increasing Sequence: A sequence where each successive term is greater than the previous term.
-
Strictly Decreasing Sequence: A sequence where each successive term is lesser than the previous term.
-
Pigeonhole Principle: A principle relevant in counting arguments that implies distribution when items exceed containers.
-
Proof by Contradiction: A logic principle where the opposite of the statement is assumed to derive a contradiction.
Examples & Applications
An example of a strictly increasing subsequence from (3, 7, 5, 1, 6) is (3, 5, 6).
An example of a strictly decreasing subsequence from (6, 4, 2) is (6, 4, 2).
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Increase is ascend, decrease is descend, select without end, that's a subsequence friend!
Stories
Once there was a garden where flowers grew tall and short. The tall flowers reached for the sun, but some were planted in a way that the shorter ones had to be skipped, creating a beautiful subsequence of heights.
Memory Tools
I.D.E.P. - Increasing, Decreasing, Elements, Pigeonhole – helps remember the main concepts.
Acronyms
ISD
Increasing Sequences
Decreasing Sequences - a shortcut to recall subsequence definitions.
Flash Cards
Glossary
- Increasing Sequence
A sequence where each term is less than the following term.
- Decreasing Sequence
A sequence where each term is greater than the following term.
- Subsequence
A sequence derived by selecting elements from another sequence without changing their order.
- Pigeonhole Principle
A mathematical principle stating that if n items are put into m containers, with n > m, then at least one container must contain more than one item.
- Proof by Contradiction
A proof that begins by assuming the opposite of what one aims to prove, and arriving at a contradiction.
Reference links
Supplementary resources to enhance your learning experience.