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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 going to explore the pigeonhole principle and how it can help us find disjoint groups among a set of people.
What exactly is the pigeonhole principle?
It states that if you have more items than containers, at least one container must hold more than one item. In our case, the items are the age sums, and the containers are the groups we create from the individuals.
So how does this relate to finding groups of people?
Great question! If we have 9 people, we can form 511 non-empty groups. If the possible age sums are limited, it guarantees there will be some groups with the same sum.
Could you give an example?
Sure! If we have selected individuals with distinct ages that sum to various totals, some groups will inevitably share age sums, given the limited range of possible sums.
How do we ensure those groups are disjoint?
If two groups contain the same person, we can remove the shared individual and still maintain the equality of their sums, creating disjoint groups.
To sum up, the pigeonhole principle helps us guarantee the existence of disjoint groups with equal age sums.
Now let’s establish the possible sums of ages among our 9 individuals.
What is the minimum sum we can have?
The minimum sum occurs when we only have one person aged 18. So that minimum is 18.
And what about the maximum sum?
The maximum is when all 9 are aged 58, giving us a total sum of 522.
So, what's the range of possible sums then?
The range is from 18 to 522. Therefore, the total range of possible sums is 505.
How does that play into our earlier discussion?
We have 511 ways to select groups from our 9 individuals, yet only 505 possible distinct sums. Thus, the pigeonhole principle guarantees at least some groups will share a sum, which leads us back to ensuring they can be disjoint.
We now understand the calculation behind our age sums before moving to the next proof step.
To conclude our topic, let’s review how removing shared members helps in creating disjoint groups.
What if the two groups are already disjoint?
If they are disjoint, there's no need for modification, and we have fulfilled our requirement.
And if they’re not disjoint?
In that case, we simply remove the common members, and the sums of both groups will still remain equal.
Can you summarize why this approach works?
Absolutely! By having more groups than sums, we are guaranteed overlaps, and by removing overlaps, we ensure that the two groups become disjoint yet maintain equal sums.
This really shows the power of the pigeonhole principle!
Indeed, it showcases the cleverness of combinatorial arguments in proofs. Always remember, when items exceed containers, overlaps are inevitable!
A principle used to deduce the existence of matching groups in sets where the items outnumber the containers.
The ages of individuals that are specifically different from one another, crucial in forming groups.
Understanding how to form subsets and ensuring they are disjoint is pivotal in achieving equal sums.
If more ages than groups there be, at least one sum you'll surely see!
Imagine a party where 9 friends can only pick cakes that sum up to the same calories. They’ll definitely need to share some items!
Pigeon + Groups = Equal Sums (PGES) - Remember that shared items lead to equal sums.
DRIVE - Distinct, Remove, Increase, Verify, Equal (steps to ensure groups remain disjoint with same sums).
Term
What is the pigeonhole principle?
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Term
What is a disjoint group?
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A concept in combinatorics asserting that if more items are put into containers than the number of containers, at least one container must hold more than one item.
A sequence derived by deleting some or no elements from another sequence without changing the order of the remaining elements.
Sets that have no elements in common.
A set derived from another set by including some or all of its elements.
Elements that are different from each other in a set.
The section analyzes the existence of disjoint groups of people having the same sum in a given range.
This section provides a mathematical proof utilizing the pigeonhole principle, demonstrating that among a selection of 9 people with ages ranging from 18 to 58, it is possible to find two disjoint groups whose ages sum to the same total. The discussion involves detailed reasoning and the application of set theory principles.
In this section, we delve into a fascinating application of the pigeonhole principle to demonstrate that, given 9 distinct ages drawn from a range of 18 to 58, we can always find two distinct groups of individuals whose combined age sums are equal. The proof begins with an understanding of the total number of non-empty subsets we can form from these 9 individuals, which amounts to 511.
This theorem illustrates the power of combinatorial reasoning in problem-solving.