Types of Sets and their Significance

Sets are essentially a collection of different items that constitute a group in mathematics. The element present in the set refers to each object in the set. When writing a set, curly brackets are utilised. As an example of a set, consider the following. Set A equals {1,2,3,4,5}.

Definition of set

The set is a clearly defined group of things in mathematics. Usually, we are using a capital letter to identify and represent sets. The elements that make up a set in set theory can be anything: people, alphabets, numbers, forms, variables, and so on.

A collection of prime numbers fewer than ten is defined, but a class of clever pupils is not. As a result, a set, A = 2, 3, 5, 7, can be used to represent a collection of prime numbers fewer than ten.

Set representation

For the representation of sets, various set notations are employed. The following are the three set notations that are used to represent sets:

• Semantic form of representation
• Roster form of representation
• Set-builder form of representation

Semantic form 

The semantic form of a set is a declaration that shows which items will make the set. Set A, for example, contains the first five odd numbers.

Roaster form

The roster notation, in which the elements of the sets are surrounded in curly brackets separated by commas, is the most prevalent way to represent sets. Set B, for example, is the collection of the first five even numbers: {2,4,6,8,10}. The order of the items of the set does not important in a roster form.

Set builder form

A specific rule or statement in the set-builder notation explains the common feature of all the elements of a set. The set builder form is represented as a vertical bar with text detailing the character of the set’s elements. A = {k | k is an even number, k ≤ 20} . According to the statement, all of the components of set A are even numbers less than or equal to 20.

Types of set

Different sorts of sets are identified. Singleton, finite, infinite, empty, and other terms are among them.

Singleton set

A singleton set, also known as a unit set, is a set with only one element. Set A = k | k is an integer between 3 and 5, resulting in A = 4.

Finite set

A finite set is one with a finite or countable number of elements, as the name implies. Set B = {k | k is a prime number smaller than 20}, for example, is B = {2,3,5,7,11,13,17,19}.

Infinite set

An infinite set is a set containing an unlimited number of items. Set C to ‘multiples of 3’ as an example.

Null or empty sets

An empty set, also known as a null set, is one that has no elements. The symbol “∅” is used to represent an empty set. It’s pronounced as ‘phi.’ Set X = {} as an example of null set.

Equal set

Equal sets are defined as two sets that contain the same elements. A = {1,2,3,5} and B = {1,2,3,5} are two examples. A = B can be used to represent this because both set have the same elements.

Unequal Sets

Unequal sets are those sets that must have a minimum of one element which is different from sets.

A ={0,1,2,3} and B = {2,3,4,5} are two examples. The given sets A and B are unequal here. A ≠ B can be used to indicate the unequal set.

Equivalent set

When two sets contain the same number of elements but distinct elements, they are said to be equivalent sets. For instance, A = {1,2,3,4} and B = {a,b,c,d}. Because n(A) = n(B) so sets A and B are equivalent to each other because both the set have same no of elements.

Overlapping set

If at least one element from set A appears in set B, then the two sets are said to overlap. A = {2,4,6} B = {4,8,10} is an example. Element 4 appears in both sets A and B in this case so set A and set B are overlapping sets.

Disjoint Sets

If there are no shared elements in both sets, they are disjoint sets. For instance, A = {1,2,3,4} and B = {5,6,7,8}. Sets A and B are disjoint in this case because not a single element is common.

Superset and subset

If every member in set A is also present in set B, set A is a subset of set B (A ⊆ B), and set B is the superset of set A(B ⊇ ).

Because all of the components in set A are present in set B, A = {1,2,3} B = {1,2,3,4,5,6}  so A⊆ B and set B is the superset of set A, as indicated by B ⊇ A.

Significance of set

Sets are used to store a collection of related things. Sets play a significant role in mathematics since they are used or referred to in almost every subject. They are necessary for the construction of more complicated mathematical structures. 

Conclusion

A set is a mathematical model for a collection of various objects; it contains elements or members, which can be any mathematical object, such as numbers, symbols, or points in space.