Cardinal number

In mathematics, a cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are in a set. The cardinal number associated with a set ⁠ A {\displaystyle A} ⁠ is generally denoted by ⁠ | A | {\displaystyle \vert A\vert } ⁠, with a vertical bar on each side, though it may also be denoted by A {\displaystyle A} , card ⁡ ( A ) , {\displaystyle \operatorname {card} (A),} or # A . {\displaystyle \#A.} Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. The cardinality of a finite set can be identified with a natural number, which can be found simply by counting its elements. For example, the sets ⁠ { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} ⁠ and ⁠ { 4 , 5 , 6 } {\displaystyle \{4,5,6\}} ⁠ both have the same cardinality 3, as evidenced by the bijection ⁠ { 1 ↦ 4 , 2 ↦ 5 , 3 ↦ 6 } {\displaystyle \{1\mapsto 4,2\mapsto 5,3\mapsto 6\}} ⁠. The behavior of cardinalities of infinite sets is more complex. For example, there exists a bijection between the set of all natural numbers ⁠ N {\displaystyle \mathbb {N} } ⁠ and the set of all rational numbers ⁠ Q {\displaystyle \mathbb {Q} } ⁠, and thus ⁠ | N | = | Q | {\displaystyle \vert \mathbb {N} \vert =\vert \mathbb {Q} \vert } ⁠ even though ⁠ N {\displaystyle \mathbb {N} } ⁠ is a proper subset of ⁠ Q {\displaystyle \mathbb {Q} } ⁠—something that cannot happen with proper subsets of finite sets. However, a fundamental theorem due...