Cardinality

The cardinality of a mathematical set is the number of elements contained in the set. It is also called the size of the set, and this size can be either finite or infinite. We usually denote it using vertical bars around the set’s name, like |X|.

Understanding the key properties of set cardinality helps reinforce the concept. Let’s now look at a few of these properties.

If sets A and B are disjoint, then n(A∪B) = n(A) + n(B).

• For any two sets A and B, to find how many elements are in A or B or both, use the formula:  n(A ∪ B) = n(A) + n(B) – n(A ∩ B). 
  This is referred to as the inclusion-exclusion principle.

• If we choose any three sets A, B, and C, the number of elements in their union is given by the formula:
  n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C).

• The relation "having the same number of elements" is an equivalence relation because it is reflexive, symmetric, and transitive.

• A set is said to be countable if the elements in it can be listed one by one, like counting with natural numbers, or if it has a finite number of elements.

• If a set cannot be counted, it is uncountable.

• Sets such as N (natural numbers), Z (integers), and Q (rational numbers) are countable.

• Real numbers, or the set R, cannot be counted.

• Any smaller part (subset) of a countable set can also be counted.

• If a set contains an uncountable set is also uncountable.

• If both A and B are countable, then their Cartesian product A × B is also countable.

A set A is called countable if it meets one of these two conditions:

• A is a finite set.

• Or, its elements can be listed one by one just like counting natural numbers, that is, there is a one-to-one correspondence with the set of natural numbers (N).

If a set is both countable and infinite, it is called a countably infinite set. Examples include the natural numbers (N), integers (Z), and rational numbers (Q).

For finite countable sets, the cardinality is simply the number of elements. For countably infinite sets, the cardinality is the same as that of the natural numbers.

If there is no one-to-one correspondence between set A and the natural numbers, then it is uncountable. One commonly used example is the set of real numbers (R).  Similarly, any interval of numbers like  [a, b] or (a, b), where a < b, is also uncountable.

It is important to note that a finite set is always countable. Uncountably infinite sets have a cardinality larger than that of the natural numbers.

The power set is the collection of all possible subsets of a set, including the empty set and the set itself. If a set A has n elements, where n is a non-negative integer, then its power set contains 2ⁿ subsets. The cardinality of the power set is always greater than that of the original set. For example, if A = {1, 2, 3, 4}, then A has 4 elements, its power set will contain 2⁴ = 16 subsets.

The number of elements that make up a set is its cardinality. For example, if A = {1, 2, 3, 4}, it contains 4 elements, so its cardinality is 4.

The cardinality of any finite set is always a natural number.
Usually, the cardinality of a set A is written as |A| or n(A). It can also be shown as card(A) or #A.

Examples:

If A = {l, m, n, o, p}, then |A| = n(A) = 5

If P = {Red, Green, Blue, White}, then |P| = n(P) = 4.

For finite sets, the cardinality is nothing but the number of elements in a set. However, for infinite sets, we have a different notation.
The cardinality of countably infinite sets is denoted by aleph-null (ℵ₀). This represents the size of a countably infinite set, like the set of natural numbers (N).

So, if set A is countable and infinite, we can say its cardinality is the same as that of natural numbers:
n(A) = n(N) = ℵ₀.

Let's look at two sets, A and B, which can be either infinite or finite. Then:

• A and B are the same size if each element in A pairs exactly with one element in B and vice versa (a one-to-one and onto match):  n(A) = n(B).

• The size of A is less than or equal to the size of B if each element in A pairs with a unique element in B, but some items in B may remain unmatched: n(A) ≤ n(B).

• A is smaller than B if each element in A pairs one-to-one with elements in B, but some elements in B are left unmatched:  n(A) < n(B).

Cardinality is an important concept that has been used in various fields beyond math. Let’s now learn how it can be applied in real life. 

• In schools, cardinality is used to keep a count of students participating in different activities. For example, if students are grouped on the basis of their extracurricular activities, cardinality gives the number of students in each group.

• This concept is widely used in inventory management to maintain a record of the number of different items in stock.

• E-commerce platforms use cardinality to track product preferences by analyzing the number of items purchased or viewed over time.