Set Cardinality — the Number of Elements of a Set

Definition 1. The cardinality of a set is its size. For afinite set, the cardinality of a set is the number of members it contains. In symbolic notation the size of a set S is written |S|. We will deal with the idea of the cardinality of an infinite set later.

The number of elements of a set
If S is a finite set, the symbol |S| stands for the number of elements of S. For example:

If S = {1, 3, 5, 7, 9}, then |S|=5.
If A = {1001, 1002, 1003, …, 3000}, then |A|=2000.
If B = {letters in the English alphabet}, then |B|=26.

The set C = {5} is a one—element set (singleton) because |C|=1. It is important to distinguish between the number 5 and the set C = {5}:

5∈C but 5≠C.

Example 1: Set cardinality
For the set D={1, 2, 3} we show cardinality by writing |D|=3.

Ex2. Finding the following cardinalities.
(a) |{x∈ℤ∶|x|<10}|=19 because {x∈ℤ∶|x|<10}={-9,-8,-7,…,0,…7,8,9}. (b) |{x∈ℤ∶x2<10}|=7 because {x∈ℤ∶x2<10}={-3,-2,-1,0,1,2,3}. (c) |{x∈ℕ∶x2<0}|=0

Definition 2. Cardinal Number
The cardinal number of set A. symbolized by n(A), is the number of elements in set A.
Size of a set. The cardinality of a set is the number of elements contained in the set and is denoted n(A).

Both set A={1,2,3} and set B={England, Brazil, Japan} have a cardinal number of 3; that is, n(A)=3, and n(B)=3. We can say that set A and set B both have a cardinality of 3.

Ex3. Let A={1,2,3,4,5} B={1,3,5} C={4,6}. Find the cardinality of the given set.
(a) n(A)
(b) n(AC)
🔑 (a) 5 (b) 6
Singletons — What is a Singleton?