Types of Sets in Set Theory

In mathematics, a Set is a fundamental concept representing a collection of well-defined objects or elements. Sets are typically denoted by capital letters, and the individual elements within a set are listed in curly braces, separated by commas. For example, A={1,2,3,4,5} represents a set A with elements 1, 2, 3, 4, and 5. The order of elements within a set does not matter, and elements are not repeated. The number of elements in a set is referred to as its cardinality.

Sets can vary widely in nature, ranging from finite sets with a limited number of elements to infinite sets that continue indefinitely. This article delves into the definitions and examples of each type, providing a comprehensive overview of sets and their significance in mathematical theory.

What is Set?

A Set is a collection of well-defined objects or elements. 
A set is represented by a capital letter. The number of elements in the finite set is known as the cardinal number of a set.

Let us take an example:

         A = {1, 2, 3, 4, 5}

Since a set is usually represented by capital letters, thus, A is the set and 1, 2, 3, 4, 5 are the elements of the set or members of the set. The elements that are written in the set can be in any order but cannot be repeated. All the set elements are represented in small letters in the case of alphabets.  Also, we can write it as 1 ∈ A, 2 ∈ A, etc. The cardinal number or the cardinality of the set A is 5.

Types of Sets

(i) Singleton Set

A set consisting of only one element is said to be a Singleton set.

For example: Set S = {5}, M = {a} are said to be singleton since they consist of only one element 5 and 'a' respectively.

(ii) Finite Set

A set whose number of elements is countable i.e. finite or a set whose cardinality is a natural number (∈ N) is said to be a Finite set.

For example: Sets
A = {a, b, c, d}, B = {5,7,9,15,78} and C = { x : x is a multiple of 3, where 0<x<100)

Here A, B and C all three contain a finite number of elements i.e. 4 in A, 5 in B and 33 in C and therefore will be called finite sets.

(iii) Infinite Set

A set containing infinite number of elements i.e. whose cardinality can not found is said to be an Infinite set. 
Thus, the set of all natural numbers.

N = {1, 2, 3, 4 . . . .}  is an infinite set.

Similarly, the set of all rational numbers between any two numbers will be infinite. For example,

A = {x : x ∈ Q, 2 < x < 5} is an infinite set. 

(iv) Equal Sets

When two sets consists of same elements, whether in the same order, they are said to be equal.
In other words, if each element of the set A is an element of the set B and each element of B is an element of A, the sets A and B are called equal, i.e., A = B. 

For example, A = {1,2,3,4,5} and B = {1,5,2,4,3} , then A = B.

(v) Empty Set

If a set consists of no element (zero elements), it is said to be the empty set. It is denoted by ∅. It is also called null set or void set. 
A common way of representing the null set is given by 

∅ = { x : x ≠ x }, this set is empty, since there is no element which is not equal to itself. For example, a = a, 2 = 2.

(vi) Subsets of a Given Set

Suppose A is a given set. Any set B, each of whose elements is also an element of A, is called contained in A and is said to be a subset of A.
The symbol ⊆ stands for "is contained in" or "is subset of". Thus, if "B is contained in A" or "B is subset of A", we write 

              B ⊆ A.

When B is subset of A, we also say 'A contains B' or 'A is superset of B.

The symbol ⊇ is  read for "contains" this A ⊇ B means "A contains B".

Example : If A = (3, 5, 7), B = (3, 5, 7, 9) than A ⊆ B since every element of A is also an element of B. But B ⊄ A since 9 ∈ B while 9 ∉ A.

(vii) Proper Subset

If B is a subset of A and B ≠ A, then B is said to be proper subset of A. In other words, if each element of B is an element of A and there is at least one element of A which is not an element of B, then B is said to be a proper subset of A. "Is proper subset of" is symbolically represented by ⊂.

Also, the empty set ∅ is a proper subset of every set except itself. 

(viii) Improper Subset

Set A is called an improper subset of B if and only if A = B. 

Note : Every set is an improper subset of itself.

(ix) Power Set

The set of all subsets of a given set A, is said to be the power set of A.

The power set of A is denoted by P(A). 

If the set A= {a, b, c} then its subsets are ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} {a, b, c}.

Therefore, P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} {a, b, c} }.

(x) Universal Set

A set which consists of all the elements of the considering sets is said to be the Universal set for those sets.
It is generally denoted by U or S.

For example : 
Consider the following sets, A = {a, b, c, d, e} ; B = {x, y, z} and U = {a, b, c, d, e, f, g, h, w, x, y, z}
Here, U is the universal set for A and B, since U contains all the elements of A and B.

Types of Sets in Set Theory- Table

Image below shows table of Sets, their symbols and examples.

Conclusion

In conclusion, sets form of mathematical theory, providing a fundamental framework for understanding and organizing collections of objects. Whether dealing with finite or infinite sets, equal or empty sets, the concept of subsets and power sets, or the universal set, each type of set brings unique properties and applications that are critical for various mathematical disciplines. By grasping these basic concepts, one can appreciate the structured yet flexible nature of sets, enabling further exploration into more advanced topics in mathematics and its related fields. Understanding sets not only enhances mathematical literacy but also builds a foundation for logical reasoning and problem-solving skills

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