At Least 2 People in a Set of n Have Same # of Acquaintances

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In summary, in a set that has n people, at least two of them will have the same number of acquaintances. This can be proven by considering the function K(a), which maps each person in the set to the number of acquaintances they have. Since there are n people in the set, but only a range of n possible values for the number of acquaintances, the function cannot be injective and therefore at least two people must have the same number of acquaintances.
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kliker
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show that in a set that has n people at least 2 of them have the same amount of acquaintances in the set

well what i tried is that first of all if we have n people one person can know n-1 people within the set so having a set of n people at least 2 of them should have the same amount of acquaintances in the set

but, i don't think its right, maybe someone could guide me please, i would appreciate it

i really think that i haven't fully understood how to use this principle here
 
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Let K(a) be the number of acquaintances of a; then, you can only say that [itex]0 \leq K\left(a\right) \leq n-1[/itex]. This is a range of n, which is exactly the number of persons, so you cannot infer (yet) that K(a)=K(b) for some a and b.

Think like this: the function K goes from the set {1,...,n} (persons) to {0,1,...,n-1} (number of acquaintances of each person); if you manage to prove that this function cannot be injective, for n > 1, then you are done.
 

1. How can you prove that at least 2 people in a set of n have the same number of acquaintances?

There are a few different ways to prove this statement, but one common approach is through the use of the Pigeonhole Principle. This principle states that if there are n+1 objects that need to be placed into n distinct groups, then at least one group must have 2 or more objects in it. In the context of this question, the "objects" are the individuals in the set and the "groups" are the number of acquaintances each person has. Since there are n+1 people and only n possible numbers of acquaintances, by the Pigeonhole Principle, at least 2 people must have the same number of acquaintances.

2. Why is it important to study the number of acquaintances in a set of people?

Understanding the number of acquaintances in a group can provide valuable insights into social networks and human behavior. It can help us understand how people form relationships, how information spreads through a group, and how individuals influence each other. Additionally, this type of research can have practical applications in fields such as marketing, public health, and disaster response, where understanding social networks can be crucial.

3. Does this statement apply to all sets of people, regardless of size?

Yes, this statement holds true for all sets of people, regardless of size. As long as there are n+1 people and only n possible numbers of acquaintances, there will always be at least 2 people with the same number of acquaintances.

4. How does the number of acquaintances a person has affect their social interactions?

The number of acquaintances a person has can greatly influence their social interactions. People with a larger number of acquaintances tend to be more socially active and are often seen as more popular. This can lead to increased social opportunities and a wider range of connections. On the other hand, people with a smaller number of acquaintances may have more meaningful and deeper relationships with those individuals.

5. Can this statement be applied to any type of relationship, or just acquaintances?

This statement specifically refers to acquaintances, as it is based on the number of connections a person has within a group. However, similar principles can be applied to other types of relationships and social networks. For example, the "Birthday Paradox" is a similar concept that states that in a group of 23 people, there is a 50% chance that 2 people will share the same birthday. This principle can be applied to other types of relationships or characteristics within a group.

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