kardinaliteettia

K cardinaliteettia, or cardinality in English, is a mathematical notion that measures the size of a set. It applies to both finite and infinite collections. For a finite set, the cardinality is the number of elements. For infinite sets, cardinality generalizes size through the existence of bijections: two sets have the same cardinality if there is a one-to-one correspondence between their elements.

The concept was introduced by Georg Cantor in the late 19th century as part of set theory. The symbol |A| denotes the cardinality of a set A. Infinite sets can have well-defined sizes that extend beyond natural counting numbers. The set of natural numbers has cardinality aleph-null (ℵ0), indicating countable infinity. The real numbers form a larger cardinality, commonly denoted by the continuum c, which equals 2^{ℵ0}.

Key properties include that two sets are equinumerous if a bijection exists between them. Cantor’s theorem shows that, for any set A, its power set P(A) has strictly greater cardinality than A, implying a strict hierarchy of infinite sizes. Basic arithmetic with cardinals follows special rules for infinite sizes: for infinite κ and λ, the sum κ+λ and the product κ·λ both equal max(κ, λ). Cardinal exponentiation κ^λ is more nuanced and can yield larger sizes, depending on the operands.

Cardinality is foundational in many areas of mathematics, including combinatorics, analysis, and logic. It provides a precise way to compare sizes of finite and infinite sets, and it underpins concepts such as injectivity, surjectivity, and function spaces. In Finnish mathematical usage, kardinaliteetti and kardinaliteettia are the terms used for these ideas.