Cardinality

Cardinality refers to the number that is obtained after counting something. Thus, the cardinality of a set is the number of elements in it. For example, the set {1, 2, 3, 4, 5} has cardinality five which is more than the cardinality of {1, 2, 3} which is three. The cardinality of countable infinite sets is equal to the cardinality of the set of natural numbers.

In this article, we will explore the concept of the cardinality of different types of sets (finite, infinite, countable and uncountable). Let us learn more about the cardinality of finite and infinite sets in detail along with a few examples for a better understanding of the concept.

The cardinality of a set is defined as the number of elements in a mathematical set. It can be finite or infinite. For example, the cardinality of the set A = {1, 2, 3, 4, 5, 6} is equal to 6 because set A has six elements. The cardinality of a set is also known as the size of the set. It is denoted by the modulus sign on both sides of the set name, |A|.

A finite set is a set with a finite number of elements and is countable. An infinite set, on the other hand, has an infinite number of elements, and an infinite set may be countable or uncountable. Yes, finite and infinite sets don't mean that countable and uncountable. There is a difference. For example, sets like N (natural numbers) and Z (integers) are countable though they are infinite because it is possible to list them. In other words, we can have a one-to-one correspondence (bijection) from each of these sets to the set of natural numbers N, and hence they are countable. On the other hand, the set of all real numbers R is uncountable as we cannot list its elements and hence there can't be a bijection from R to N.

Cardinality of Countable Sets

To be precise a set A is called countable if one of the following conditions is satisfied.

• A is a finite set.
• If there can be a one-to-one correspondence from A → N. i.e., n(A) = n(N).
  (This point is used to determine whether an infinite set is countable.)

If a set is countable and infinite then it is called a "countably infinite set". Some examples of such sets are N, Z, and Q (rational numbers). So, the cardinality of a finite countable set is the number of elements in the set. On the other hand, if it is an infinite countable set, then its cardinality is equal to the cardinality of the set of natural numbers.

Cardinality of Uncountable Sets

A set A is said to be uncountable (or) "uncountably infinite" if they are NOT countable. The set of real numbers is an example of uncountable sets. Similarly, intervals like [a, b], (a, b], [a, b), (a, b) (where a < b) are also uncountable sets. Remember that a finite set is never uncountable. The cardinality of uncountable infinite sets is either ℵ₁ or greater than this.

Power set of a set is the set of all subsets of the given set. If a set A has n elements, then the cardinality of its power set is equal to 2ⁿ which is the number of subsets of the set A. In general, we can say that the cardinality of a power set is greater than the cardinality of the given set. If a set A = {1, 2, 3, 4}, then the cardinality of the power set of A is 2⁴ = 16 as the set A has cardinality 4.

The cardinality of a set is nothing but the number of elements in it. For example, the set A = {2, 4, 6, 8} has 4 elements and its cardinality is 4. Thus, the cardinality of a finite set is a natural number always. The cardinality of a set A is denoted by |A|, n(A), card(A), (or) #A. But the most common representations are |A| and n(A). Here are some examples:

• If A = {a, b, c, d, e}, then n(A) (or) |A| = 5
• If P = {Sun, Mon, Tue, Wed, Thu, Fri, Sat}, then n(P) (or) |P| = 7

As we have already seen in the first section, the cardinality of a finite set is just the number of elements in it. But the cardinality of a countable infinite set (by its definition mentioned above) is n(N) and we use a letter from the Hebrew language called "aleph null" which is denoted by ℵ₀ (it is used to represent the smallest infinite number) to denote n(N). i.e., if A is a countable infinite set then its cardinality is, n(A) = n(N) = ℵ₀. An uncountable set always has a cardinality that is greater than ℵ₀ and they have different representations. For example, the cardinality of the uncountable set, the set of real numbers R, (which is a lowercase "c" in Fraktur script). To summarize:

• The cardinality of any countable infinite set is ℵ₀.
• The cardinality of an uncountable set is greater than ℵ₀.

Let us consider two sets A and B (finite or infinite). Then:

• n(A) = n(B) if there can be a bijection (both one-one and onto) from A → B.
• n(A) ≤ n(B) if there can be an injection (one to one and maybe onto) from A → B.
• n(A) < n(B) if there can be an injection (only one-one but strictly not onto) from A → B.

For point 3, the best example is n(N) < n(R) (i.e., the cardinality of the set of natural numbers is strictly less than that of real numbers as N is countable and R is uncountable).

Now that we know the meaning of the cardinality of a set, let us go through some of its important properties which help in understanding the concept in a better way.

• If A and B are two disjoint sets, then n(A U B) = n(A) + n (B).
• For any two sets A and B, n (A U B) = n(A) + n (B) - n (A ∩ B). This is popularly known as the "inclusion-exclusion principle".
• For any three sets A, B, and C, n(A U B U C) = n (A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n (A ∩ B ∩ C).
• The relation of sets having the same cardinality is an equivalence relation.
• A set A is countable if it is either finite or there is a bijection from A to N.
• A set is uncountable if it is not countable.
• The sets N, Z, and Q are countable.
• The set R is uncountable.
• Any subset of a countable set is countable.
• Any superset of an uncountable set is uncountable.
• If A and B are countable then their cartesian product A X B is also countable.

Important Notes on Cardinality

• The cardinality of a set is the number of elements in the set.
• The cardinality of any countable infinite set is ℵ₀.
• The cardinality of an uncountable set is greater than ℵ₀.

☛ Related Topics:

• Cardinal Numbers
• Ordinal Numbers
• Types of Sets
• Sets Formulas

What is the Meaning of Cardinality in Math?

The cardinality of a set means the number of elements in it. For any set A, its cardinality is denoted by n(A) or |A|. But for infinite sets:

• The cardinality is ℵ₀ if the set is countably infinite.
• The cardinality is greater than ℵ₀ if the set is uncountable.

Here, ℵ₀ is called "Aleph null" and it represents the smallest infinite number.

How to Find Cardinality of a Finite Set?

The cardinality of a set A is denoted by n(A) and is different for finite and infinite sets.

• If A is finite, then n(A) is the number of elements in A.
• If A is countably infinite, then n(A) = ℵ₀.
• If A is uncountable, then n(A) > ℵ₀.

How to Find Cardinality of an Infinite Set?

There are two types of infinite sets: countable and uncountable. The cardinality of an infinite set that is countable is ℵ₀ whereas the cardinality of an infinite set that is uncountable is greater than ℵ₀.

What is Uncountability?

A set is said to be uncountable if its elements cannot be listed. The set of all real numbers is an example of an uncountable set. In other words, there can't be a bijection from the set of real numbers to the set of natural numbers.

Is the Cardinality of an Uncountable Set Infinity?

No, the cardinality can never be infinity.

• If the set is infinite and countable, its cardinality is ℵ₀ (it is called aleph null and is used to represent the smallest infinite number).
• If the set is infinite and uncountable then its cardinality is strictly greater than ℵ₀.

Is the Cardinality of a Finite Set Always Finite?

Yes, the cardinality of a finite set A (which is represented by n(A) or |A|) is always finite as it is equal to the number of elements of A. For example, if A = {x, y, z} (finite set) then n(A) = 3, which is a finite number.

What are the Formulas Related to Cardinality?

For any three sets A, B, and C:

• n (A U B) = n(A) + n (B) - n (A ∩ B)
• n(A U B U C) = n (A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n (A ∩ B ∩ C).

What is the Cardinality of a Power Set of a Finite Set?

The cardinality of a power set of a finite set is equal to the number of subsets of the given set. So, if a finite set A has n elements, then the cardinality of its power set is equal to 2ⁿ.

How to Write Cardinality of a Set?

The cardinality of a set A is written as |A| or n(A) or #A which denote the number of elements in the set A.