Question

Show that every infinite set \(A\) contains a countably infinite set, i.e., an infinite sequence of distinct terms.

Short answer

Every infinite set contains a countably infinite subset through infinite selection of distinct elements.

Step-by-step solution

Understand the Definitions

To solve this problem, we need to understand the definitions of an infinite set and a countably infinite set. An infinite set is a set with no finite number of elements. A set is countably infinite if its elements can be put in one-to-one correspondence with the natural numbers.

Assume Existence of Infinite Set

Let's assume that we have an infinite set \( A \). Our goal is to show that there exists within \( A \) a subset that is countably infinite.

Construct a Countable Sequence

Start with any element \( a_1 \) from \( A \). Since \( A \) is infinite, there exists at least one more element, so choose another element \( a_2 \) that is distinct from \( a_1 \).

Continue the Selection Process

Since \( A \) is infinite, we can continue this process, selecting \( a_3, a_4, \ldots \) such that each \( a_n \) is distinct from the previously selected elements in the sequence.

Define the Infinite Subset

The subset \( \{ a_1, a_2, a_3, \ldots \} \) is countably infinite because you can establish a one-to-one correspondence with the natural numbers \( 1, 2, 3, \ldots \).

Conclusion

Since we can always find such a sequence in \( A \) by this method, every infinite set contains a countably infinite subset.

Key concepts

Countably Infinite Set

A countably infinite set is a fascinating concept in mathematics. It's a set that is infinite like the set of natural numbers. However, it shares a particular property that allows for each element of the set to be matched perfectly with a unique natural number. This matching is called being in "one-to-one correspondence."

Think of it this way: if you can start listing the elements of a set, and theoretically, never stop because there’s always a "next" element, you might be dealing with an infinite set. But to be countably infinite, you should also be able to keep a natural number glued to each of those elements consistently.

• The set of natural numbers itself, \(\mathbb{N} = \{ 1, 2, 3, \ldots \}\), is the quintessential example of a countably infinite set.
• If you manage to list each element in an infinite set with each element labeled by a natural number, your set is countably infinite. This can extend to sets like integers or rational numbers.

When you hear "countably infinite," it emphasizes how each element of this infinite collection can still fit along the infinite, yet orderly line of natural numbers.

One-to-One Correspondence

The idea of one-to-one correspondence, or bijection, is key to understanding countable sets. It's about pairing elements from one set with elements of another set in such a way that every element in each set has a unique partner in the other set.

Picture this: You have two sets of cards, each set having distinct, unique cards. If you lay them side by side where each card from the first set matches with exactly one card from the second, and vice versa, you've established a one-to-one correspondence.

• This matching is essential for proving that a set is countably infinite. By aligning each element to a unique natural number, you demonstrate that the set behaves like the natural numbers in its infiniteness.
• Mathematically, a function \( f \) is a one-to-one correspondence between sets \( A \) and \( B \) if \( f: A \to B \) is bijective, meaning both injective (no two elements map to the same element) and surjective (all elements in \( B \) are mapped).

Achieving a one-to-one correspondence can seem magical, providing an orderly framework to handle infinite chaos. It’s like giving all infinite stars in the sky a designated spot in a constellation.

Natural Numbers

Natural numbers are the foundation of mathematics and everyday counting. Starting from 1 and proceeding to 2, 3, and so on, they form the simplest and most familiar infinite set. These numbers, sometimes represented by the symbol \(\mathbb{N}\), are integral to various mathematical concepts, including being the benchmark for defining "countably infinite."

The natural numbers obey the following characteristics:

• They are the positive integers \(\{ 1, 2, 3, \ldots \}\).
• They start from 1, continuing without any upper bound, symbolizing the idea of infinity in a structured format.
• Each number has a unique successor, ensuring that counting never stops.

Connecting sets to the natural numbers by establishing a one-to-one correspondence allows mathematicians to dissect and understand larger sets. The ability to map infinite elements onto this orderly sequence enables a clear vision even amidst the boundless stretch of infinity.