samakardinaliteettisia

Samakardinaliteettisia is a Finnish mathematical term used to describe a relation or property of sets that have the same cardinality. The word combines sama, meaning the same, with kardinaliteetti, cardinality. In practice, two sets A and B are referred to as samakardinaliteettisia when there exists a bijection between them, which is equivalent to saying that their cardinalities are equal: |A| = |B|. The concept is central to set theory and is applicable to both finite and infinite sets.

Examples help illustrate the idea. The sets {1, 2, 3} and {a, b, c} are samakardinaliteettisia, since both have cardinality 3. In contrast, {1, 2} and {a, b, c} are not samakardinaliteettisia because their cardinalities differ. For infinite sets, the notion also applies: the natural numbers N and the even numbers 2N are samakardinaliteettisia, as there is a bijection (for example, f(n) = 2n) between them.

Properties and context: Being samakardinaliteettisia is an equivalence relation on the class of sets, meaning it is reflexive, symmetric, and transitive. It partitions sets into classes by cardinality. The concept is closely related to fundamental results such as the Cantor–Schröder–Bernstein theorem, which states that if there are injections in both directions between two sets, then they have the same cardinality.

Usage note: In Finnish mathematical writing, samakardinaliteettisia is used adjectivally, for example, "kaksi joukkoa ovat samakardinaliteettisia." Related terms include cardinality and bijection, and the theorem concepts underpinning equal cardinalities.