Uncountable sets are infinite sets that cannot be put into a one-to-one correspondence with the set of natural numbers, meaning their cardinality is strictly larger than that of countable sets. The most famous example of an uncountable set is the set of real numbers, which demonstrates that not all infinities are equal in size.

Uncountable Sets

Uncountable sets

Infinite sets are collections of elements that have no end, meaning they cannot be counted or exhausted by enumeration. They are characterized by their ability to be put into a one-to-one correspondence with a proper subset of themselves, exemplified by the set of natural numbers.

infinite sets

One-to-one correspondence, also known as bijection, is a fundamental concept in mathematics where each element in one set is paired with exactly one element in another set, and vice versa, ensuring both sets have the same cardinality. This concept is crucial for comparing the sizes of infinite sets and is foundational in defining equivalence relations and functions in set theory.

one-to-one correspondence

set of natural numbers

Cardinality refers to the measure of the 'number of elements' in a set, which can be finite or infinite, and is crucial in understanding the size and comparison of sets in mathematics. It plays a fundamental role in set theory, enabling mathematicians to distinguish between different types of infinities and to explore properties of sets in various mathematical contexts.

cardinality

A countable set is a set that can be put into a one-to-one correspondence with the natural numbers, meaning it can be enumerated or listed in a sequence. Countable sets include finite sets and countably infinite sets, and this concept is fundamental in understanding the different sizes of infinity in set theory.

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