kertalukumäärä

Kertalukumäärä, often translated as "cardinality" in English, is a fundamental concept in set theory representing the number of elements in a set. It essentially answers the question "how many?". For finite sets, the cardinality is a non-negative integer. For example, the set {apple, banana, cherry} has a cardinality of 3. The cardinality of the empty set, which contains no elements, is 0.

The concept extends to infinite sets, where cardinalities are not represented by simple integers but by transfinite numbers. Georg Cantor developed the theory of transfinite cardinalities, demonstrating that there are different "sizes" of infinity. The cardinality of the set of natural numbers (0, 1, 2, ...) is denoted by aleph-null (ℵ₀) and is the smallest infinite cardinality. The cardinality of the set of real numbers is larger than ℵ₀, a fact proven by Cantor's diagonal argument.

Cardinality is crucial in understanding set operations. For example, the cardinality of the union of two disjoint sets is the sum of their individual cardinalities. Similarly, the cardinality of the Cartesian product of two sets is the product of their cardinalities. The notation for cardinality is often vertical bars, such as |A| for the cardinality of set A.

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