cardinality

Thân thiện

A teacher writes the cardinality of a set on the chalkboard.

Definition

Noun: 1. (Mathematics) The number of elements in a set or group, considered as a property of that grouping. It is a measure of a set's "size."

Usage

• Cardinality is a fundamental concept in set theory used to compare the sizes of sets, including infinite sets.
• It is typically used with the preposition "of" (e.g., "the cardinality of a set").
• For finite sets, the cardinality is simply a natural number (e.g., 0, 1, 2, 3...).

Examples

• The set A = {apple, banana, cherry} has a cardinality of 3.
• Determining the cardinality of infinite sets, like the set of all integers, requires more advanced mathematical reasoning.
• If two sets can be paired element-for-element (a one-to-one correspondence), they have the same cardinality.

Advanced Usage

• Countable vs. Uncountable Infinity: In set theory, cardinality distinguishes between different "sizes" of infinity. The set of natural numbers has a countable infinite cardinality (denoted ℵ₀, aleph-null), while the set of real numbers has a larger, uncountable infinite cardinality.
• Cardinality of the Power Set: For any set S, the cardinality of its power set (the set of all subsets of S) is strictly greater than the cardinality of S itself.

Variants and Related Words

• Cardinal (Adjective): Of primary importance; fundamental. (e.g., ). In mathematics, a cardinal number is a generalization of a natural number used to describe the cardinality of a set.
• Cardinal Number: A number (such as 1, 2, 3) used in counting to indicate quantity, as opposed to an ordinal number (1st, 2nd, 3rd) which indicates position.

Synonyms

• Size (of a set)
• Magnitude (in the specific context of set theory)

Related Concepts (Not Phrasal Verbs/Idioms)

• One-to-One Correspondence (Bijection): A function pairing each element of one set with exactly one element of another set, used to define equal cardinalities.
• Set: A well-defined collection of distinct objects.
• Finite Set: A set with a cardinality that is a natural number.
• Infinite Set: A set that is not finite.

Học thuật

Noun

1. (mathematics) the number of elements in a set or group (considered as a property of that grouping)