Question

Infinite Pigeon-Hole Principle: Suppose \(X\) is an infinite set and \(Y\) is a finite set. Now, suppose \(F: X \rightarrow Y\). Show there is a \(y \in Y\) such that for infinitely many \(x \in X\) such that \(F(x)=y\).

Short answer

There exists a \(y \in Y\) with infinitely many \(x \in X\) such that \(F(x) = y\).

Step-by-step solution

Understanding the Principle

The Infinite Pigeon-Hole Principle states that if you have infinitely many objects (domain \(X\)) and a finite number of boxes (codomain \(Y\)), at least one box must contain infinitely many objects. In this exercise, \(F: X \rightarrow Y\) is a function mapping objects from an infinite domain \(X\) to a finite codomain \(Y\).

Identifying the Elements of the Problem

Notice that \(Y\) is finite with elements \( \{y_1, y_2, ..., y_n\} \). Each element in \(X\) is mapped by \(F\) to one of the elements in \(Y\). This means that the set of pre-images under \(F\) for each \(y_i\) can either be finite or infinite.

Applying the Infinite Pigeon-Hole Principle

Since \(X\) is infinite, and \(Y\) is finite, the pre-images \(F^{-1}(y_i)\) for at least one \(y_i \in Y\) must contain infinitely many elements of \(X\). This follows from the principle since we cannot distribute all infinite\(x\) into finite\(y\) without one of the \(y\) having an infinite count allocated.

Concluding the Argument

We have shown that for at least one \(y \in Y\), \(F^{-1}(y)\) is infinite. This means there exists \(y \in Y\) such that for infinitely many \(x \in X\), \(F(x) = y\). This conclusion is derived directly from the structure of the Infinite Pigeon-Hole Principle when mapping an infinite set to a finite set.

Key concepts

Infinite Sets

An infinite set is a collection of elements that extends indefinitely. Unlike finite sets, infinite sets do not have a fixed number of elements.
Examples of infinite sets include the set of all natural numbers, \( \mathbb{N} = \{1, 2, 3, ...\} \), and the set of real numbers, \( \mathbb{R} \).
These sets are boundless in size, meaning you'll never run out of elements no matter how many you count.

• Infinite sets can be of different types, such as countably infinite or uncountable.
• Countably infinite sets, like the natural numbers, can be paired one-to-one with the set of integers.
• Uncountable sets, like the real numbers, are larger and cannot be matched one-to-one with integers.

The concept of infinite sets is central to the Infinite Pigeon-Hole Principle. In this context, it implies that when assigning elements from an infinite set to categories, at least one category must receive infinitely many elements. This characteristic of infinite sets underlies the core reasoning of this principle.

Finite Sets

Finite sets are collections of elements with a definite, countable number of members. Unlike infinite sets, you can list all the elements in a finite set.
For example, consider the set of days in a week, \( \{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday\} \).
This set is finite because it contains exactly seven elements.

• Finite sets can be fully described since they have a manageable number of elements.
• They are often simpler to work with mathematically because they do not stretch to infinity.
• In the context of the Infinite Pigeon-Hole Principle, finite sets serve as the 'boxes' for sorting elements from an infinite set.

When mapping elements from an infinite set into a finite set, we cannot fit an infinite quantity into finite containers without at least one container becoming overloaded with an infinite number of items. Understanding finite sets is crucial to grasp why the Infinite Pigeon-Hole Principle holds true.

Function Mapping

In mathematics, function mapping is the process where each element from one set (known as the domain) is paired with an element in another set (the codomain).
If we consider a function \( F: X \rightarrow Y \), it means that each element \( x \) from the set \( X \) is associated with an element \( y \) in the set \( Y \).
The ornate arrow \( \rightarrow \) signifies this direction of association, or mapping.

• Function mappings can vary, being injective, surjective, or bijective.
• Injective (or one-to-one) mappings assign unique elements of the domain to each element of the codomain but not necessarily all of them.
• Surjective (or onto) mappings ensure every element of the codomain corresponds to at least one element from the domain.
• Bijective mappings are both injective and surjective, providing a perfect match between domain and codomain elements.

In reference to the Infinite Pigeon-Hole Principle, the function mapping from an infinite set to a finite set highlights that some elements in the codomain must be mapped by multiple, potentially infinite, elements from the domain. Thus, this mapping concept is vital for applying the principle when solving problems involving infinite and finite sets.