What Is a Set?

Set Theory is the part of mathematics that deals with a group of things referred to as numbers or set components. A combination of elements contained in curly brackets is referred to as a set. In other words, we can define the set as a collection of distinct objects that number, alphabets or any different quantity. Each number or alphabet is called an element in the set. These elements of the set can be subdivided into smaller groups known as subsets. In Sets, the order isn’t particularly crucial, which means 1, 5, 4 is the same as 4,5, 1. By the end, you will know what is a set and the representation of sets.

What Is A Set?

Sets are nothing but a collection of different products that combine together to constitute a group of elements in mathematics. An element of the set is the term used for each object in the set. Writing a set requires the use of curly brackets to the fullest. A set can carry any number of elements, such as letters, numbers, days of the holidays in a year, types of bikes, and so on. Set A consists of the numbers A = {1,2,3,4,5}. A set of various objects can be represented in several methods and notations. A set builder form or a roaster form are generally used to represent a more detailed form in the sets. We will take a detailed glance at each of these terms. The elements that make up a set in set theory can be anything: humans, letters of the alphabet, shapes, numbers, variables, and so on. Examples of Sets Even numbers less than 20, i.e., 2, 4, 6, 8, 10, 12, 14, 16, 18. Prime Factors of 18 are 2, 3, 6, 9. Types of Triangles on the basis of sides:  Isosceles, Equilateral, Scalene Top two doctors in the U.K. 10 Famous Engineers of the U.K.

Cardinal Number Of A Set

The cardinal number denotes the total number of items in a set, also called cardinality or order of the set. For even numbers less than 20, n(A) = 9.  {2, 4, 6, 8, 10, 12, 14, 16, 18}. So the cardinality of this set is 9. One of the most critical requirements for defining a set is that all of its elements be related to one another and share a standard feature. For example, if the elements of a set are the names of days in a month, we can claim that each of the set’s elements is the days of the year.

Basic Formulas Of Set

For any two sets P and Q, n(P U Q) = n(P) + n(Q) – n(P ∩ Q) n (P ∩ Q) = n(P) + n(Q) – n(P U Q) n(P) = n(P U Q) + n(P ∩ Q) – n(Q) n(Q) = n(P U Q) + n(P ∩ Q) – n(P) n(P – Q) = n(P U Q) – n(Q) n(P – Q) = n(P) – n(P ∩ Q) For any two disjoint sets P and Q, n(P U Q) = n(P) + n(Q) P ∩ Q = ∅ n(P – Q) = n(P)

Representation Of Sets

Curly brackets are used to indicate the sets. For example, {1,5,19} or {p,q,r} or {Days, Months, Years}, for example. The elements of the sets are represented in one of three ways: Statement, Roster, or Set Builder.

Statement Form(Semantic Form)

The elements of the set are provided with a well-defined explanation in this representation. For example, consider the following instances. If we want to write a set of prime numbers less than or equal to 20. In statement form, we can write {prime numbers less than or equal to 20}.

Roaster Form(Tabular Form)

The roster notation, where the elements of the sets are contained in curly brackets bisected by commas, is the most popular way to represent sets. Set B, for example, is the collection of the five odd numbers less than 11: 2,4,6,8,10. The order of the numbers in the set does not affect a roster form; for example, the set of the five odd numbers less than 11 can also be written as 2,8,6,10,4. Also, if a set contains an infinite number of elements, they are defined by consecutive dots at the end of the last element. Infinite sets, for example, are written as A = 1, 3, 5, 7 …, where A is the set of consecutive positive prime numbers.

Set-Builder Form

There is a rule or a declaration in the set-builder notation that defines the common trait of all the components of a set. The set builder form is represented as a vertical bar with notes telling the character of the set’s elements. For example, A = { p | p is an odd number, k ≤ 10}. This logical statement explains everything that all the elements of set A are odd numbers less than or equal to 10. Sometimes a colon (:) is also used in the place of the vertical bar (|).

Conclusion

The number of items in a set determines what type of set it is. Sets are groups of components that are of the same kind. For instance, a collection of positive integers, natural numbers, and so on. In this article, you got to learn what a set is, detailed information of the sets in maths, types of representation of sets is, and the examples and the types of sets. Set is a topic that can be applied both in the field of mathematics as well as in the field of logic.