Definition 1. The

cardinalityof 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 setSis 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∶*x*^{2}<10}|=7 because {*x*∈ℤ∶*x*^{2}<10}={-3,-2,-1,0,1,2,3}.
(c) |{*x*∈ℕ∶*x*^{2}<0}|=0

Definition 2.Cardinal Number

Thecardinal numberof setA. symbolized by n(A), is the number of elements in setA.

Size of a set. Thecardinalityof 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(*A*⋃*C*)

🔑 (a) 5 (b) 6

Singletons — What is a Singleton?