A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or the same number of instances, is observed in a variety of present-day animal … See more

In the above section, "cardinality" of a set was defined functionally. In other words, it was not defined as a specific object itself. However, such an object can be defined as follows.
The relation of having the same cardinality is called See more

Our intuition gained from finite sets breaks down when dealing with infinite sets. In the late nineteenth century Georg Cantor, Gottlob Frege, Richard Dedekind and others rejected the view that the whole cannot be the same size as the part. One example of this is See more

If A and B are disjoint sets, then
From this, one can show that in general, the cardinalities of unions and intersections are related by the following equation: See more

While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite).
Definition 1: |A| = |B| See more

If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions:
• Any set X with cardinality less than that of the See more

• If X = {a, b, c} and Y = {apples, oranges, peaches}, where a, b, and c are distinct, then | X | = | Y | because { (a, apples), (b, oranges), (c, peaches)} is a bijection between the sets X … See more