Types Of Sets

A set is an unordered collection of objects known as set elements or set members. A set A element ‘a’ can be written as ‘a A’, ‘a A’ denotes that an is not an element of the set A. 

A set can be represented in a variety of ways. In this representation, a well-defined description of the factors of the set is given. Below are some examples of the same.

1. The set of all even quantities is much less than 10.

2. The set of the wide variety is much less than 10 and more than 1. 

Knowing More On Types Of Sets

A set is a mathematical representation of a collection of different items;  A set is made up of elements or members, which can be numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. 

Types Of Sets:

1. Finite Set : 

A finite set is a collection of elements with a fixed number of elements.

Example − S = { x | x ∈ N and 80 > x > 60 }

2. Infinite Set :

 An infinite set is a collection of components with an endless number of possibilities.

Example − S = { x | x ∈ N and x > 20 }

3. Subset:

An infinite file is a set of components with an endless number of possibilities.

Example  − Let, X = { 1, 2, 3, 4, 5, 6 } and Y = { 1, 2 }. Here set Y is a subset of set X as all the elements of set Y are in set X. Hence, we can write Y ⊆ X.

4. Proper Subset:

The term “proper subset” can be described as “subset of however no longer equal to”. A Set X is a suited subset of set Y (Written as X ⊂ Y ) if every issue of X is an aspect of set Y and $|X| < |Y|. 

Example − Let, X = { 1, 2, 3, 4, 5, 6 } and Y = { 1, 2 }. Here set Y ⊂ X since all elements in X are contained in X too and X has at least one element is more than set Y.

5. Universal Set:

It is a collection of all factors in a unique context or application. All the units in that context or software are surely subsets of this frequent set. Universal units are represented as U. 

Example − We can also define U as the set of all animals on earth. In this case, the set of all mammals is a subset of U, the set of all fishes is a subset of U, the set of all insects is a subset of U, and so on. 

6. Empty Set : 

An empty set contains no elements. It is denoted by using ∅. As the variety of elements in an empty set is finite, the empty set is a finite set. The cardinality of an empty set or null set is zero.

Example − S = { x | x ∈ N and 7 < x < 8 } = ∅

7. Unit Set:

Singleton set or unit set contains only one element. A singleton set is denoted by { s }.

Example − S = { x | x ∈ N, 7 < x < 9 } = { 8 }

8. Equal Set: 

If two sets contain the same elements they are said to be equal.

Example − If A = { 1, 2, 6 } and B = { 6, 1, two }, they are equal as every factor of set A is an element of set B and each component of set B is a thing of set A. 

9. Equivalent Set: 

If the cardinalities of two sets are the same, they are called equivalent sets.

Example − If A = { 1, 2, 6 } and B = { 16, 17, 22 }, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A| = |B| = 3

10. Disjoint Set:

Two units A and B are known as disjoint sets if they do not have even one issue in common.

n(A ∩ B) = ∅

n(A ∪ B) = n(A) + n(B)

Example − Let, A = { 1, 2, 6 } and B = { 7, 9, 14 }, there is not a single common element, hence these sets are overlapping sets.

Conclusion:

In this course, we have learned that modern algebra is a study of sets with operations defined on them. As the main example, we have started a systematic study of groups. Group theory is one of the most important areas of contemporary mathematics, with applications ranging from physics and chemistry to coding and cryptography. It is also one of the research interests in this school. Further study of groups can be undertaken in the appropriate honors modules.

Cantor concluded that the units N and E have equal cardinality. Cantor then proved that there is no one-to-one correspondence between the set of actual numbers.