Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
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.
Understanding Increasing and Decreasing Subsequences
Unlock Audio Lesson
Today, we're going to explore subsequences in sequences of distinct real numbers. Who can tell me what a strictly increasing sequence looks like?
Isn't it a sequence where each number is less than the following one?
Exactly! It's a sequence like (1, 2, 3). Now what about a strictly decreasing sequence?
That's where each number is greater than the one before, right?
Yes! For example, (5, 4, 3). It's essential to understand these definitions since today's focus is on subsequences of a sequence with n + 1 distinct real numbers.
The Concept of Subsequences
Unlock Audio Lesson
Now let’s talk about subsequences. Can anyone define a subsequence?
I think it’s a sequence derived from another sequence by selecting some elements without changing their order but not necessarily using all of them.
Great definition! For example, from the sequence (1, 3, 0, -5, 2, 8), (1, 2, 8) is a subsequence.
So we can skip numbers to form a subsequence?
Correct! Remember this when we discuss how we can always find a specific length, n + 1, of increasing or decreasing subsequences.
Applying the Pigeonhole Principle
Unlock Audio Lesson
Now, let's dive into how we prove our main statement using the pigeonhole principle. Who can remind us what this principle states?
It says that if you have more pigeons than holes, at least one hole must contain more than one pigeon.
Exactly! In our case, if you have n + 1 distinct numbers, we want to find pairs of lengths in increasing or decreasing subsequences.
So, we could have pairs of subsequence lengths that match because there's a repetition given more choices than lengths?
Yes! This leads us to a contradiction if we assume all sizes are bounded by n.
Finding Length of Subsequences
Unlock Audio Lesson
Let's look closer at how we can identify lengths of subsequences. If I define lengths for subsequences starting at each element, what do you think happens?
The lengths must vary, right? Some can have increasing lengths while others decrease.
That's right! And based on our previous discussions, the pigeonhole principle will help us show that one of those lengths must be greater than n.
It’s like a form of contradiction where you assume limits and find they've been surpassed.
Exactly! You've grasped the key concept we’re proving today.
Summarizing Key Concepts
Unlock Audio Lesson
Let’s summarize! What have we learned about strictly increasing and decreasing subsequences?
We’ve learned how to define and find finer lengths of subsequences.
And we used the pigeonhole principle to show that with n + 1 numbers, we are guaranteed a length of n + 1.
Correct! Remember these insights as they lay the groundwork for complex mathematical proofs in the future.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The discussion elaborates on the characteristics of strictly increasing and strictly decreasing sequences formed from any sequence of distinct real numbers. It explains how the pigeonhole principle is applied to ensure the existence of a subsequence of length more than one that satisfies these conditions.
Detailed
Detailed Summary of Conclusion on Divisibility
In this section, we investigate properties of subsequences within sequences of distinct real numbers. Specifically, for any sequence of n + 1 distinct real numbers, there exists at least one subsequence of length n + 1 that is either strictly increasing or strictly decreasing.
A strictly increasing sequence is defined such that each number is less than its successor, while a strictly decreasing sequence is where each number exceeds its successor. Key to this conclusion is the application of the pigeonhole principle, which asserts that if we consider the lengths of the longest increasing and decreasing subsequences following any element in the sequence, we can derive contradictions that show at least one subsequence of length n + 1 must exist.
The section delves into concrete examples and the techniques of proof by contradiction, leading us to realize the power of structure in seemingly arbitrary sequences and further deepening our understanding of sequence behavior in mathematics.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Understanding Subsequences
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
We want to show that if you take any sequence of n + 1 distinct real numbers, then there always exists a subsequence of length n + 1 that is either strictly increasing or strictly decreasing.
Detailed Explanation
The main idea revolves around finding subsequences in a given sequence of distinct real numbers. A strictly increasing subsequence is a sequence of elements where each element is greater than the one before it. Conversely, a strictly decreasing subsequence is where each element is less than the one before it. The challenge is to prove that no matter how we choose these n + 1 numbers, one of these subsequences (either increasing or decreasing) will always exist.
Examples & Analogies
Think of a group of friends whose heights vary. If you take a group of 8 friends (which is n + 1, where n = 7) and you want to find a way to stand them in a line such that their heights are either all increasing or all decreasing, you will always be able to arrange at least some of them in this manner because there are just enough distinct heights that force this situation due to the nature of how numbers work.
Defining Increasing and Decreasing Subsequences
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
A strictly increasing sequence is of the form (a₁, a₂, …) where a₁ < a₂ < a₃ < ... < aₘ, while a strictly decreasing sequence is of the form (a₁, a₂, …) where a₁ > a₂ > a₃ > ... > aₘ.
Detailed Explanation
Here, we clarify what strictly increasing and strictly decreasing sequences are. An increasing sequence means that as we move through the sequence from the first element to the last, each next element is larger than the previous. On the other hand, in a decreasing sequence, each next element is smaller than the previous one. Understanding these definitions is crucial for the following steps in the proof.
Examples & Analogies
Consider a set of stairs. If you keep stepping up, you're moving in an increasing manner. If you're stepping down, that's like a decreasing sequence. Just like you can't stand on the same step twice if you're going up or down, the elements in these mathematical sequences must be distinct.
The Pigeonhole Principle
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
To prove our claim, we use the pigeonhole principle along with proof by contradiction. Assume that for every number in the sequence, the length of the longest increasing subsequence is at most n.
Detailed Explanation
The pigeonhole principle is a common inference tool which states that if you have more items than containers, at least one container must contain more than one item. In our case, if we assume that every sequence can hold only a certain amount of longest subsequences, we eventually get a contradiction when we try to pair these subsequences. Since we have more distinct elements than the capacity of subsequences, we conclude that at least one subsequence must exceed the assumed maximum length.
Examples & Analogies
Imagine you have 24 pairs of socks but only 23 drawers to store them in. According to the pigeonhole principle, at least one drawer must contain more than one pair of socks. This represents how we cannot have more sequences of length at most n than there are unique numbers available.
Applying Contradiction
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Assume that there exists some numbers that contradict the maximum length assumption. If m is the number of pairs found, we can rearrange them such that we have a longer subsequence.
Detailed Explanation
By assuming the opposite of the claim (that no subsequence can be longer than n), we show that this leads to an impossibility in how the pairs are structured. When we rearrange them under the earlier defined relations of increasing or decreasing subsequences, we eventually find that at least one subsequence must necessarily break the initial assumption, therefore confirming the claim.
Examples & Analogies
Think of a group of students trying to arrange their chairs in a line. If you have more students than chairs, a couple of students must step out of line to fit into the arrangement without sitting in the same chair. Thus, if each divided group must have some overlapping seating, it leads you to realize that at least one student must reappear in a different sequence.
Conclusion and Generalization
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
The existence of either an increasing or decreasing subsequence of length n + 1 is guaranteed by the properties laid out thus far.
Detailed Explanation
In conclusion, by utilizing definitions of strictly increasing and decreasing sequences along with the pigeonhole principle, we assert that any selection of distinct real numbers will inherently contain either an increasing or decreasing arrangement of length n + 1. This universality in behavior reinforces the fundamental structures within number theory and sequences.
Examples & Analogies
This conclusion parallels common experiences. For instance, if you have a set of unique colors and you try to arrange them in a sequence without repeating, eventually, you'll find a similar pattern of color arrangements that either transition smoothly from light to dark or vice-versa, due to the nature of distinct yet relatable shades.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Strictly Increasing Sequence: A sequence where terms consecutively increase.
-
Strictly Decreasing Sequence: A sequence where terms consecutively decrease.
-
Subsequence: A selection of elements from a sequence that maintains the order.
-
Pigeonhole Principle: A principle stating that more items than containers will result in at least one container with multiple items.
-
Proof by Contradiction: A rationale demonstrating the falsity of an assumption.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
In the sequence (2, 4, 6), you have a strictly increasing subsequence, while in (8, 5, 3), you have a strictly decreasing subsequence.
-
From the sequence (1, -2, 3, 5, -1), you can form subsequences like (1, 3, 5) which is strictly increasing.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
If the numbers rise, it's increasing, no surprise! If they fall, it's decreasing, it's very wise!
📖 Fascinating Stories
-
Imagine climbing a staircase (increasing) versus sliding down a slide (decreasing). Each showcases sequences in a fun way!
🧠 Other Memory Gems
-
For subsequences, remember 'SOS' - Select, Order, Skip - to form your sequence!
🎯 Super Acronyms
SIDS for Strictly Increasing and Decreasing Sequences
- S: for Strictly
- I: for Increasing
- D: for Decreasing.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Strictly Increasing Sequence
Definition:
A sequence where each term is greater than the previous one.
-
Term: Strictly Decreasing Sequence
Definition:
A sequence where each term is less than the previous one.
-
Term: Subsequence
Definition:
A derived sequence where elements are chosen from another sequence without reordering, and some may be skipped.
-
Term: Pigeonhole Principle
Definition:
A principle in mathematics stating that if more items are put into fewer containers, then at least one container must contain more than one item.
-
Term: Proof by Contradiction
Definition:
A method of proof where you assume a statement is false and show that this leads to a contradiction.