Cardinality is an important concept in set theory and mathematics due to its general applications and importance across various occupations. Thus, the cardinality of a set is the number of elements in it along with examples. Cardinality is important in every field such as cryptocurrency, markets, etc.
For example, the set {1, 2, 3, 4, 5} has a cardinality of 5.
In this article, we will know what the cardinality of a set is. Cardinality refers to the number that is obtained after counting something.
What is Set?
Set is a collection of distinct objects, considered as a whole. These objects are called the elements or members of the set.
For example, the set of natural numbers less than 5 can be written as {1, 2, 3, 4}.
Cardinality of a Set
The number of elements in the set or a measure of its size is known as the cardinality of a set. It can be finite or infinite.
For example, the set A = {1, 2, 4} has a cardinality of 3 because it has three elements.
Cardinality of any set A is denoted by: |A| or card(A)
Examples of Cardinality of a Set
Some other examples includes:
- 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
Cardinality of Different Sets
Some of the common sets with cardinality are:
- Cardinality of a Power Set
- Cardinality of a Finite Set
- Cardinality of a Infinite Set
- Cardinality of Cartesian Products
Cardinality of a Finite Set
Cardinality of a finite set refers to the number of elements in the set. If a set S is finite, its cardinality is simply the count of distinct elements within the set.
The total numbers are in the set is known as the cardinality of a power set.
For example: If A = {1, 2, 3, 4, 5}, then |A| = 5.
Cardinality of Infinite Set
A set is countably infinite if its elements can be put into a one-to-one correspondence with the natural numbers N = {1, 2, 3, . . . }. This means that you can list the elements of the set in a sequence (even if the sequence goes on forever).
The cardinality of a countably infinite set is denoted by ℵ0.
Examples:
- Natural Numbers: The set of natural numbers N={1, 2, 3, . . .} is countably infinite.
- Integers: The set of integers Z={. . . ,−2, −1, 0, 1, 2, . . . } is countably infinite because you can list them in a sequence like 0, 1, −1, 2, −2, . . .
- Rational Numbers: The set of rational numbers Q = {a/b ∣ a,b ∈ Z, b ≠ 0} is countably infinite, though it's less obvious. The rationals can be arranged in a sequence by arranging fractions by their sum of numerator and denominator.
Cardinality of a Power Set
Power Set of a set S is the set of all possible subsets of S, including the empty set and S itself.
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.
If a set S has n elements, the power set P(S) will have 2n elements. This is because each element in S can either be included in or excluded from a subset, leading to 2n possible subsets.
For any finite set S with n elements: ∣P(S)∣ = 2n
Consider the set S = {a, b}.
- Subsets of S are:
- {} (the empty set)
- {a}
- {b}
- {a, b}
- The power set P(S) is {{}, {a}, {b}, {a, b}}
Since S has 2 elements, the cardinality of the power set P(S) is 22 = 4.
Cardinality of Cartesian Products
The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
Finite Set Example: If A = {1, 2} and B = {x, y, z}, then A × B = {(1, x), (1, y), (1, z), (2, x), (2, y), (2, z)}, so ∣A × B∣ = 6.
General Formula: If ∣A∣=m and ∣B∣ = n, then ∣A × B∣ = m × n.
Formulas Related to Cardinality
- 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.
Application
Some of the common application of cardinality are:
- Cardinality is used in database management to optimize queries by determining the number of unique records in a dataset.
- Cardinality is used in designing and optimizing databases.
- Cardinality is used to compare the size of input sets and the complexity of algorithms.
- Cardinality plays a crucial role in cryptography, especially in the design of cryptographic keys.
Solved Questions: Cardinality of a Set
Q1: Let A={1, 2, 3, 4, 5}. What is the cardinality of set A?
Solution:
The cardinality of set A is the number of elements in the set, i.e., ∣A∣=5
Q2: Given two sets B = {a, b, c} and C = {d, e, f, g}, What is the cardinality of the union B∪C?
Solution:
B∪C = {a, b, c, d, e, f, g}
The cardinality of B∪C is i.e., ∣B∪C∣ = 7
Q3: Let D = {2, 4, 6} and E = {1, 2, 3, 4, 5, 6}. Find the cardinality of D∩E.
Solution:
D∩E = {2, 4, 6}
The cardinality of D∩E is:
∣D∩E∣=3
Q4: Consider the set F = {x ∣ x is an integer and −3 ≤ x ≤ 3}. What is the cardinality of set F?
Solution:
F = {−3, −2, −1, 0, 1, 2, 3}
The cardinality of F is ∣F∣ = 7
Q5: Let G = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}. Find the cardinality of set G.
Solution:
The cardinality of set G is:
∣G∣=7
Q6: Suppose H = {} is an empty set. What is the cardinality of set H?
Solution:
The cardinality of the empty set is always zero i.e., ∣H∣=0
Practice Problems: Cardinality of a Set
Problem: For each Q below, determine the cardinality of the given set.
Q 1: I = {10, 20, 30, 40}.
Q 2: J={a, b, c, d, e}.
Q 3: K={x ∣ x is an even number between 1 and 10}.
Q 4: L={z ∣ z is a vowel in the English alphabet}.
Q 5: M={n ∣ n is a prime number less than 10}.
Q 6: N={r, s, t, u, v, w, x, y, z}.
Q 7: O={p ∣ p is a positive integer less than 4}.
Q 8: P = {}.
Q 9: Q={1, 1, 2, 2, 3, 3}.
Q 10: R = {y ∣ y is an integer and −2 ≤ y ≤ 2}.
Answer key
1: ∣I∣=4
2: ∣J∣=5
3: ∣K∣=4
4: ∣L∣=5
5: ∣M∣=4 (Prime numbers less than 10 are 2, 3, 5, 7)
6: ∣N∣=9
7: ∣O∣=3 (Positive integers less than 4 are 1, 2, 3)
8: ∣P∣=0 (Empty set)
9: ∣Q∣=3 (Unique elements are 1, 2, 3)
10: ∣R∣=5 (Set includes integers -2, -1, 0, 1, 2)
Read More,
Conclusion
Cardinality is a fundamental idea in set theory, illustrating the size of a set, i.e., the number of elements it contains. It is useful in various fields, such as mathematics, computer science, and even real-life systems like grouping or categorizing items. Understanding how to compute the cardinality of a set allows for more useful data management, resource funding, and decision-making procedures, making it a useful tool in both academic and used contexts.
Similar Reads
Cardinality in DBMS In database management, cardinality represents the number of times an entity of an entity set participates in a relationship set. Or we can say that the cardinality of a relationship is the number of tuples (rows) in a relationship. It is a fundamental concept that helps define how data in different
4 min read
Complement of a Set In mathematics, a set is a collection or grouping of well-defined objects. All such objects, when grouped in a set, are called elements. Sets are represented by capital letter symbols, and the elements are placed together in a curly bracket {}.For example, if W is the set of whole numbers, then W =
10 min read
Cantor Set Cantor set is a basic notion in mathematics that plays an extremely important role in mathematical areas such as set theory and topology. It was first developed by the German mathematician Catarus at the end of the 19th century. Cantor Set exemplifies most intriguingly how seemingly straightforward
13 min read
Equal Sets: Definition, Cardinality, and Venn Diagram In mathematics, a set is a collection of well-defined and distinct objects. These objects are called the elements of the set. The elements of a set can be listed in any order, but repetition is not allowed. Sets are typically represented by capital letters, such as A, B, etc. Sets can be empty, sing
9 min read
Finite Sets Finite set is a collection of finite, well-defined elements. For better understanding, imagine you have a bunch of your favourite toys or snacks. You know exactly how many you have, that's the idea of a finite set in math. A finite set is a way to discuss collections of things you can count. In this
10 min read