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A short class with Sarah and her students. The other ways are under More.
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Today, we're diving into subsequences. A subsequence can be derived from another sequence while maintaining the original order, even if they are not consecutive. For instance, in the sequence 1, 3, 0, -5, 2, 8, we can form a subsequence like 1, -5, 2.
So, does that mean any selection from a sequence, maintaining the original order, is a subsequence?
Exactly! And remember, subsequences can be strictly increasing or decreasing. Can anyone provide an example of each type?
For a strictly increasing example, we could use 1, 3, 5. And for decreasing, maybe something like 5, 3, 1?
Great examples! Remember that ordering matters. We'll explore more about these subsequences and how we use them in broader mathematical proofs.
Now, let’s discuss the pigeonhole principle. This principle tells us that if we have more items than containers, at least one container must contain more than one item.
Can you give an example of that, maybe in everyday life?
Sure! Imagine you have 10 pairs of shoes but only 9 slots in your closet. At least one slot will have two pairs of shoes. In our case, we will apply it to our subsequences to prove their existence.
How does that relate to our earlier discussion on subsequences?
It’s instrumental because it helps demonstrate that within our sequence of distinct real numbers, there must exist subsequences of sufficient length that are either increasing or decreasing.
Let’s apply this to a practical example. Consider 9 people whose ages range from 18 to 58. Using the theories we've discussed, can we claim there are disjoint groups with the same sum of ages?
I think using the pigeonhole principle again might help! We can form various subsets of those 9 individuals.
Correct! And since there are 511 non-empty subsets and the sum can only range from 18 to 522, we ensure that at least two of those groups must yield the same total.
What if there are overlaps in the groups? Doesn’t that affect the result?
Good question! Even if the groups overlap, if they share common individuals, we can remove those to create completely disjoint groups while preserving the sum equalities.
A subsequence is defined as a sequence derived from another sequence where elements may not be consecutive but retain the original order. An example is taking a subset from a sequence like 1, 3, 0, -5, 2, 8, and forming a subsequence such as 3, 2, 8. The same principle applies to increasing and decreasing sequences.
We refer to a sequence of numbers that successively increases or decreases. The lengths of such subsequences can be denoted as L(i) for increasing and D(i) for decreasing, representing the length starting at index i.
This concept is crucial in proving the existence of either subsequence lengths greater than or equal to n + 1. By considering pairs of indices from the arbitrary sequence, we use the pigeonhole principle to demonstrate that at least one subsequence must exist that is either strictly increasing or strictly decreasing.
The section concludes with an application of these principles to a tangible problem, where we can always form two disjoint groups of people from a set of nine individuals whose ages are distinct and within a given range, proving that their sum can be the same.
In an order you'll hide, a subsequence will ride, some numbers will abide, while others set aside.
Imagine a group of friends at a party each holding distinct ages between 18 and 58, and how they can pair up based on shared interests but also ensuring no two can be from the same age.
To remember the steps for finding subsequences: S-Select, M-Maintain order, and J-Judge elements.
SPICY: Sequential Pairs In a Check for Yields (subsequences and sums).
Term
Subsequence
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Term
Pigeonhole Principle
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A sequence derived from another sequence by retaining the original order but allowing for non-consecutive elements.
A sequence where each element is greater than the previous, maintaining order without repeats.
A principle that states if more items are put into fewer containers, then at least one container must contain more than one item.
Groups that do not overlap, meaning no elements are shared between the two groups.
This section discusses the establishment of disjoint groups of distinct real numbers ensuring subsequences that are either strictly increasing or decreasing.
The section explains the concept of subsequences within an arbitrary set of distinct real numbers and uses the pigeonhole principle to demonstrate the existence of either a strictly increasing or strictly decreasing subsequence. It practically applies these concepts to illustrate how to organize distinct ages into groups with the same sum.
This section focuses on demonstrating that within any sequence of n + 1 distinct real numbers, we can always find a subsequence of length n + 1 that is either strictly increasing or strictly decreasing.
Subsequences: A subsequence is defined as a sequence derived from another sequence where elements may not be consecutive but retain the original order. An example is taking a subset from a sequence like 1, 3, 0, -5, 2, 8, and forming a subsequence such as 3, 2, 8. The same principle applies to increasing and decreasing sequences.
Long Increasing/Decreasing Subsequences: We refer to a sequence of numbers that successively increases or decreases. The lengths of such subsequences can be denoted as L(i) for increasing and D(i) for decreasing, representing the length starting at index i.
Pigeonhole Principle: This concept is crucial in proving the existence of either subsequence lengths greater than or equal to n + 1. By considering pairs of indices from the arbitrary sequence, we use the pigeonhole principle to demonstrate that at least one subsequence must exist that is either strictly increasing or strictly decreasing.
The section concludes with an application of these principles to a tangible problem, where we can always form two disjoint groups of people from a set of nine individuals whose ages are distinct and within a given range, proving that their sum can be the same.