The empty set has cardinality 0. We write this as \(|\emptyset| = 0.\)
For any whole number \(N \geq 1\) define \(A_N = \{1,2,\dots,N\}\). For example, \(A_2 = \{1,2\}\) and \(A_5 = \{1,2,3,4,5\}.\)
The cardinality of a non-empty set \(A\) is \(|A| = N\) if there exists a bijection \(f: A_N \rightarrow A.\)
Example: Let \(A = \{a,b,c\}.\) Then \(|A| = 3\) because there is a bijection \(f: A_3 \rightarrow A.\) \[f(1) = a, f(2) = b, f(3) = c\] In other words, the elements of \(A\) can be numbered \(1,2,3.\)
Cardinality of a power set of a finite set.
Cardinality of an infinite set.
Countable versus uncountable infinities.