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A short class with Sarah and her students. The other ways are under More.
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Today, we will explore sequences. Can anyone tell me what a strictly increasing sequence is?
Isn't it a sequence where each term is larger than the previous one?
Exactly! For example, (1, 2, 3) is a strictly increasing sequence. Now, how about a strictly decreasing sequence?
That's one where each term is smaller than the one before it, like (3, 2, 1).
Well done! Keep in mind these definitions as they lead to understanding the Pigeonhole Principle.
Next, let's talk about subsequences. Who can define what a subsequence is?
Is that a sequence derived from another sequence where you can skip some terms?
Correct! For example, from the sequence (1, 3, 5), we can form (1, 5) as a subsequence.
But it doesn't have to be in order, right?
No, it doesn't have to be consecutive. Now, let’s use this concept with the Pigeonhole Principle.
We're proving that in any sequence of n + 1 distinct numbers, there is a subsequence of length n + 1 that is either strictly increasing or strictly decreasing. Can anyone explain how we can start?
We can define the lengths of the longest increasing and decreasing subsequences for each number?
Exactly! Let’s denote them as 'L' for increasing and 'D' for decreasing.
And if L and D are both at most n, we would have n distinct pairs.
Correct. But since we have n + 1 numbers, the Pigeonhole Principle ensures we will find overlapping values in these pairs, leading us to a contradiction.
In our proof by contradiction, we assumed L and D could not exceed n, leading to a necessary overlap. What does this imply?
It means that there must be a number where the increasing or decreasing subsequence is longer than n.
Correct! Hence, we conclude there must exist a subsequence of length n + 1 that is either strictly increasing or strictly decreasing, confirming our initial claim.
This really highlights the practical applications of the Pigeonhole Principle!
A fundamental principle used to prove the existence of certain structures within sets.
A sequence formed where each element is greater than those preceding it.
A sequence where each element is less than those preceding it.
A sequence derived from another sequence where order is maintained but some elements can be omitted.
If it’s strictly growing, watch it fly, each number’s higher, oh me, oh my!
Imagine climbing a staircase; you can only go up to get higher! Just like numbers in a sequence must rise!
For subsequences, remember 'SKIP and KEEP' – you skip but keep the order!
SISD: Strictly Increasing, Strictly Decreasing for remembering types of sequences.
Term
Pigeonhole Principle
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Term
Subsequence
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Term
Strictly Increasing Sequence
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A sequence where each term is greater than its predecessor.
A sequence in which each term is less than its predecessor.
A sequence derived from another sequence by deleting some elements without altering the order of the remaining elements.
A mathematical principle stating that if n items are put into m containers, with n > m, then at least one container must hold more than one item.
This section discusses the Pigeonhole Principle and its application in proving the existence of increasing or decreasing subsequences in a sequence of distinct real numbers.
The section centers around the Pigeonhole Principle, which asserts that in 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. This is demonstrated via a proof by contradiction using lengths of decreasing and increasing subsequences.
This section delves into the core of the Pigeonhole Principle by establishing that, in any sequence containing n + 1 distinct real numbers, there exists a subsequence of length n + 1 that is either strictly increasing or strictly decreasing. This assertion is validated through a proof by contradiction. The initial definitions are set for increasing and decreasing sequences, followed by an explanation of subsequences, which can skip values from the original sequence. By introducing variables for the lengths of these subsequences, the paradox arises when assuming that the length of all increasing and decreasing subsequences never exceeds n. Ultimately, using the Pigeonhole Principle, it is shown that at least one subsequence must exceed this length, thereby confirming the existence of a required subsequence.