Kardinalitu

Kardinalitu, also known as cardinality, is a fundamental concept in set theory, a branch of mathematical logic. It refers to the size or number of elements in a set. Cardinality is used to compare the sizes of different sets, even if those sets are infinite.

The cardinality of a set is denoted by the symbol |S|, where S is the set. For finite sets, the cardinality is simply the number of elements in the set. For example, the set {1, 2, 3} has a cardinality of 3.

Infinite sets can also have cardinality. The smallest infinite cardinality is aleph-null (ℵ0), which is the cardinality of the set of natural numbers (1, 2, 3, ...). Other infinite cardinalities exist, such as ℵ1, ℵ2, and so on, each representing a larger infinity.

Two sets have the same cardinality if there is a one-to-one correspondence between their elements. This means that each element in one set can be paired with exactly one element in the other set, and vice versa. For example, the set of natural numbers and the set of even numbers both have the same cardinality of ℵ0, because there is a one-to-one correspondence between them (1 ↔ 2, 2 ↔ 4, 3 ↔ 6, ...).

Cardinality is a crucial concept in mathematics, as it allows for the comparison and classification of sets based on their size, regardless of whether they are finite or infinite. It is also used in various areas of mathematics, including topology, analysis, and algebra.