Cardinality Of Sets

We have the idea that cardinality should be the number of elements in a set. This works for sets with finitely many elements, but fails for sets with infinitely many elements. We approach cardinality in a way that works for all sets. First we define when we consider two sets to have the same cardinality.

The number of elements in a set is the cardinality of that set. The cardinality of the set 92A92 is often notated as 92A92 or 92nA92. Note a set can be finite or infinite. A finite set will have a cardinality of 0 or a natural number. An infinite set has a cardinality of the form 9292aleph_092 aleph null, which represents the

2 Cardinality 2.1 'Same cardinality' 2.1.1 Denition 2.1 We say that sets X and Y have the same cardinality if there exists a bijection f X! Y. We express this symbolically by writing jX jjY j. Note that in Denition 2.2 we do not dene the cardinality, jX j, of a set X. 2.2 'Not greater cardinality' 2.2.1

Theorem . Any subset of a countable set is countable. Any superset of an uncountable set is uncountable. Proof. The intuition behind this theorem is the following If a set is countable, then any quotsmallerquot set should also be countable, so a subset of a countable set should be countable as well.

A bijection, comparing a set of apples to a set of oranges, showing they have the same cardinality. In mathematics, the cardinality of a finite set is the number of its elements, and is therefore a measure of size of the set. Since the discovery by Georg Cantor, in the late 19th century, of different sizes of infinite sets, the term cardinality was coined for generalizing to infinite sets the

Thus, the cardinality of a finite and countable set is the number of elements in that set. Meanwhile, the cardinality of an infinite and countable set is the cardinality of the set of natural numbers. Now, let us understand the cardinality of the set B 4, 8, 12, 16, with the set of natural numbers .

Learn what cardinality of a set means, how to find it, and how to compare it with other sets. Explore the concepts of finite, infinite, countable, and uncountable sets with examples and diagrams.

Find the cardinality of set G. Solution The cardinality of set G is G7. Q6 Suppose H is an empty set. What is the cardinality of set H? Solution The cardinality of the empty set is always zero i.e., H0. Practice Problems Cardinality of a Set. P roblem For each Q below, determine the cardinality of the given set.

The cardinality of a set is nothing but the number of elements in it. For example, the set A 2, 4, 6, 8 has 4 elements and its cardinality is 4. Thus, the cardinality of a finite set is a natural number always. The cardinality of a set A is denoted by A, nA, cardA, or A. But the most common representations are A and nA.

Learn the definition and examples of cardinality, a measure of a set's size. Find out how to compare the cardinality of finite and infinite sets, and the difference between countable and uncountable sets.