Cardinality of sets
Equal cardinality
We say that two sets \( X \) and \( Y \) have equal cardinality iff there exists a bijection \( f : X \rightarrow Y \) from \( X \) to \( Y \).
Cardinality n
Let \( n \) be a natural number. A set \( X \) is said to have cardinality n if it has equal cardinality with the set \( \{i \in \mathbb{N} : 1 \ge i \ge n \} \). We also say that such a set has \( n \) elements.
Finite sets
A set is finite iff it has cardinality \( n \) for some natural number \( n \); otherwise, the set is called infinite.
Notation: if \( X \) is a finite set, we use \( \#(X) \) to denote the cardinality of \( X \).
Example
The set of natural numbers is infinite.
Sets with equal cardinality can have one contain the other. For example, there is a bijection between the set of natural numbers and the set of even natural numbers, \( f: X \rightarrow Y\), \( f(n) := 2n \).