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A short class with Sarah and her students. The other ways are under More.
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Today, we’ll start by discussing subsequences! Who can tell me what a strictly increasing sequence looks like?
Is it like when each number is larger than the previous one?
Exactly! An increasing subsequence will look like (a1, a2, ..., an) where a1 < a2 < ... < an. How about a strictly decreasing sequence?
It should have each number smaller than the one before it!
Correct! An example would be (b1, b2, ..., bm) where b1 > b2 > ... > bm. Remember, subsequences can skip some elements!
Wait, how do we find these subsequences in a set of numbers?
Great question! We’ll see how the pigeonhole principle helps us assure the existence of such subsequences regardless of the chosen numbers.
So, each number in a sequence can have its own increasing or decreasing length?
Exactly! That's essential for our proof!
To summarize, we learned about increasing and decreasing subsequences. Next, we’ll apply the pigeonhole principle for our proof.
Now let’s dive into how we can demonstrate the existence of these subsequences using the pigeonhole principle. What do we mean by that?
Is that where we have more objects than containers?
Exactly! If you have more pigeons than holes, at least one hole must contain more than one pigeon. Here, we’re dealing with lengths of subsequences!
So we assign lengths to each number?
Right! We denote Li as the length of the longest increasing subsequence starting at ai. If we assume every length is ≤ n, we can explore pairs (Li, Di) as potential values.
This sounds complex! How do we derive a contradiction?
Good query! If we assume that all lengths are less than or equal to n, we'd end up with more pairs than possible values. Hence, we have to have the same lengths for two different starting points.
And that means we can find one subsequence that’s longer!
Exactly! Summarizing today, we’ve seen how the pigeonhole principle helps us find increasing or decreasing subsequences. Now, let’s look at an illustrative example using an age group.
For our final segment, let’s see how we can apply our findings to ages of individuals. What limitations did we set for a group of 9 people?
Ages from 18 to 58!
Correct! So if we pick any 9 people, how many possible non-empty subsets can be formed?
511! Since there are 2^9 subsets minus the empty set.
Exactly! Now, what about the sum of these ages?
The minimum is 18 and the maximum is 522 if everyone is 58!
Perfect! Now, applying the pigeonhole principle, we have 511 subsets as our pigeons and a limited range of sums as our holes. What does that guarantee?
That there must be at least two groups of people whose age sums are equal!
Exactly! And even if any groups share members, we can modify them to ensure they’re disjoint. Great job today everyone! To summarize, we’ve seen the principles of increasing and decreasing subsequences with an impactful real-world application.
A selection of elements from a sequence where order is preserved.
A combinatorial argument that ensures that if there are more items than containers, at least one container must hold more than one item.
A sequence with each term larger than the previous term; crucial for proving the existence of specific subsequences.
A sequence where each term is less than the previous; also essential for the proof.
In sequences you’ll find, numbers can align, increasing or dropping like leaves in the pine.
Imagine a classroom of distinct students, each standing in a line. Some are taller, some shorter. If there are more students than heights available, there must be a repeat in a height category!
To remember increasing subsequences, think 'Ladder': Each rung gets higher, never lower.
SIPS - Subsequence Increases and Pigeonhole Sequences.
Term
What is a subsequence?
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Term
Describe the pigeonhole principle.
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A sequence where each element is larger than the one preceding it.
A sequence where each element is smaller than the one preceding it.
A sequence derived from another sequence by deleting some or no elements without changing the order of the remaining elements.
A principle stating that if more items are put into fewer containers than there are items, at least one container must hold more than one item.
A logical inconsistency, where a statement contradicts another; often used in proofs.
This section explores how to apply the pigeonhole principle to demonstrate the existence of increasing or decreasing subsequences within any selected group of distinct real numbers.
In this section, we discuss the application of the pigeonhole principle in proving that for any selection of distinct real numbers, there will always exist a subsequence that is either strictly increasing or strictly decreasing. It includes examples and explores specific cases, such as the grouping of ages within given limits, and highlights the significance of both increasing and decreasing subsequences.
In this section, we begin with a statement that for any sequence comprised of (n + 1) distinct real numbers, there always exists a subsequence of length (n + 1) that is either strictly increasing or strictly decreasing. Concepts of subsequences are clarified, highlighting that these are not required to be consecutive. The proof employs the pigeonhole principle and contradiction, leading to the conclusion that some elements must share properties that force a subsequence of the required nature to exist. An illustrative example is provided using an age group of people, demonstrating the method of forming disjoint groups to arrive at equal sums of ages. This highlights the essence of the pigeonhole principle in real-world applications.