Sets and Their Representation

Introduction:

The study of sets includes the collection of objects and proper knowledge of the set. The theory of sets further provides information on the diverse sets related to their representation and symbol, including the varieties of operations performed.

Definition of sets

In mathematics, sets are the genuine collection of objects that can be further represented in a set-builder or can also include the roster form with various sets. Normally, sets are denoted within the curly braces {}. For instance, A= {5, 6, 7, 8}, this is the number that forms up the set. 

However, in simple terms, elements of a set or set itself can be defined as the collection of well-defined, organized elements or objects that will not change from person to person. However, the objects can include anything such as the likes, age, dislike, entities that rely on the simple number system to the complex number system. However, there is a thing that is highly noticeable while forming the sets of objects and entities include:

• A family of all-natural and whole numbers can be presented in the set, including even numbers, odd numbers, rational numbers, real numbers, and integers. 
• It can also have the collection of names of the Indian soldiers. 
• Collection of night and day temperature can also be present in the elements of a set.

• Various collective data that are gathered from the MOM, and ISRO.
• Group of possible outcomes that are available from the coin toss and the dice roll. 

Apart from the above things, several other things can be included in the sets. Also, another important thing to consider is that the rule of set further can define the nature that helps to collect a single or group of objects in the universal platform. 

What is the Element of Sets?

To understand the elements of a set, you can further take an example that includes, 

A= {6, 7, 8, 9}

Therefore, a set is denoted with the help of a letter that is a capital letter. Thus, it is concluded from the above example that A represents the set, and 6, 7, 8, 9 represent the elements of a set. This further can be in any order, but repetition is not allowed in the sets. Furthermore, the small case letter alphabets can be used for letter representation. For instance, 1e A, 2e A, etc. Some of the common terms found in the sets are:

• Z represents the Set of Integers.
• N denotes the Natural Numbers.
• Q represents the set of all rational numbers.
• Z* denotes the positive numbers.
• R denotes the set of real numbers. 

Orders of Set

The order of sets further defines the numbers present in the elements of a set, which can also include the size of a set. This is further also commonly known as cardinality. In addition, for a finite set, the set will occur in a finite order, and for an infinite set, the order will become infinite.

Representation of the Elements of a set

The representation of the sets is further defined as the following:

• Roster Form- The roaster form of elements in a set can include all the elements of the particular set. 

• Statement Form- This is the form of sets that includes the curly brackets from the closer and enclosure of a set of numbers or objects. 

• Set Builder Form- To represent the set builder form of a set, Venn diagrams are the best and simple way for representation. 

What are the various types of Sets? 

The various types of sets are further provided:

Singleton Sets- This is a single set with one element present in it. This further means that if A is the set, then A = {8}, this is an example of the Singleton Set. 

Empty Sets- This Set is a set that contains no element in it. This is also considered as the Null or Void set of elements. Thus, {} is the form that can be represented in these types of sets. 

For instance: Let Set A = {a: a} this is the number of students available both in classes 6 and 7, respectively. Thus, it is obvious that a single person cannot learn in both classes. Thus it can be said that A is an Empty Set of elements. 

Another example of such Sets, is set B = {a: 1< a< 2,}, here a is representing the natural number. Also, we know that natural numbers cannot come in decimal form. Thus, set B can be represented in the form of Sets.

Finite and Infinite Set- The set with a finite set of numbers includes all the finite numbers or elements. However, the set whose elements cannot be estimated, still containing some number or figure that is larger than its precise form of a set, is defined as the infinite set. 

For instance, for such sets, let’s assume A = {3, 4, 5, 6, 7}. All the integers present in the set form this type of set of numbers or elements. 

Another example of Infinite set includes:

Set C = {Number of cows in Australia}. This represents an infinite set of numbers because the numbers are not provided and can be considered approximate. Due to this, the actual number of cows cannot be determined. 

Subsets- A set S is considered as the Subsets of Set T only when the element of the set S can belong from the set T. This can also guarantee that the element of set S is also available in the set T. Thus a subset of a set can be defined by the symbol (⊂) and can also be denoted as S ⊂ T.

This can also be written as the S ⊂ T if p ∊ S ⇒ p ∊ T. 

In the above notation, it is found that S is the subset of T, where P is one element of both S and T. 

Equal Sets- If the element or number of set A is equal to set B, then these sets form an equal set of elements. This further means that A= B, which means both have the common amount and number of elements present in it. 

For instance: Let A = {6, 7, 8, 9, 10} and B= {10, 9, 8, 7, 6}. Thus the elements present in both A and B are the same, so it can be said that both are forming an Equal set.

Universal Set- The set that can help to denote certain relevant conditions is defined as the Universal Set. For instance:

Let A= {5,6,7}, and B= (6,7,8,9,10}, from here the universal set that we can find includes, U = {5,6,7,8,9,10} 

Power Sets- A power set is defined as the set or group of all subsets for any given set, including the empty set. 

For instance set Y= {5,6}, thus we can also write:

• {} is a subset of {5, 6}. 
• {5} is the subset of {5,6}, 
• {5,6} is the subset of {5,6}
• {6} is also the subset of {5, 6}. 

Thus the power set of Y = {5, 6}, where P(Y) = ({}, {5}, {6}, {5, 6}}

Disjoint Sets- When the two varieties of the set do not hold any common elements and intersect at 0, then set X and Y are defined as the disjoint sets, which can be reported as X ∩ Y = 0.

Conclusion

These are some brief studies about the elements of a set along with its various dimensions in mathematics. This is the base topic of the set and mathematics syllabus of the IIT-JEE. This topic has a huge impact on the IIT-JEE syllabus due to its variable scope and uses in other topics. Furthermore, in this article, you can also avail of the representation of the sets in a vast form, including the various types of sets as well. The definitions of the set are also broadly classified in this article.