Set Cardinality | Discrete Math | CompSciLib

Set Cardinality

Learn about Set Cardinality

In set theory, the cardinality of a set is the total number of elements in the set (sometimes also called the size of the set). The cardinality of a set is commonly denoted using the absolute value symbol, where S represents the set:

∣ S ∣

For example, the set: A = {1, 2, 3, 4, 5} has a cardinality of 5. We can formally write this as ∣ A ∣ = 5

The Empty Set

The empty set is a set with no elements. It is denoted as:

\emptyset = {}

. The cardinality of the empty set is 0. You can also present this as :

∣ \emptyset ∣ = 0

Cardinality is important because it enables us to compare the sizes of multiple sets. It also helps us formally define the idea of finite and infinite sets. A set is finite if it has a finite cardinality, meaning it has a finite number of elements. Likewise, a set is infinite if it has an infinite number of elements, such as the set of integers:

Z = {\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots}

To find the cardinality of a set, simply count how many elements appear. For example, take the set:

A = {c, s, l, i, b}

After counting all the elements in the set, the size of the set is 5. We can formally write this as:

∣ A ∣ = 5

1) Problem: What is the cardinality of the set A = {a, b, c} ?

The cardinality of set A is 3. We can formally write this as:

∣ A ∣ = 3

Answer: 3

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