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, which can be 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:
   
  \(n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)\)
  
  This is referred to as the inclusion-exclusion principle. This formula is known 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, the 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, like counting natural numbers. In other words, 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 numerical interval, such as [a, b] or (a, b) with a < b, is uncountable. Mathematically, if a set A has n elements, then its power set has 2ⁿ elements.

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).

Here are some of the tips and tricks to master cardinality and its applications:
 

1. Start with small and simple sets. Count elements in sets like {1,2,3} or {a,b,c,d}. Practice union (A∪B) and intersection (A∩B) to see how cardinality changes. Always double-check for duplicate elements; they don’t count in sets.
    

2. Learn to use formulas for combined sets. For union of two sets; ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣. Disjoint sets: if no elements in common, just add: ∣A∪B∣=∣A∣+∣B∣. Draw Venn diagrams – they make intersections and unions easy to visualize.
    

3. Try to visualize infinite sets. Make number lines for 𝑍 or 𝑄 and grid diagrams for 𝑄 (Cantor’s pairing method). It helps us in understanding why some infinities are bigger than others.
    

4. Relate cardinality to some basic real-life examples. 
   
   Students → “number of books in library” = finite
   
   Infinite examples → “all natural numbers” or “all points on a line segment”
   
   Linking abstract ideas to tangible examples boosts memory.
    

5. Learn the difference between countable and uncountable sets:

   Countable: 𝑁, 𝑍, 𝑄
   
   Uncountable: 𝑅, interval [0,1]
   
   Use one-to-one mapping to compare sizes. If every element of A can be paired with one in B → same cardinality.

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. 

1. 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.
    

2. Computer science & programming uses cardinality for counting unique users on a website. Cardinality helps there by providing sets that are used to store unique items. It helps optimize algorithms that handle large datasets. Example: Hash tables, unique element detection, database optimization.
    

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

4. Networking & social media uses it for counting friends, followers, or connections. Cardinality helps there by understanding network size, detecting unique connections or duplicates. It uses graph theory, network analysis, and recommendation systems.
    

5. Databases and records uses cardinality. For example: A school database stores student names. Cardinality helps hereby ensuring unique records (like student IDs) → prevents duplicates. It also helps us count total students, teachers, or courses. Here, we use Data analysis, reporting, and queries in software.