Representation of Sets: The concept of sets serves as a fundamental part of present-day mathematics. In the present era, the concept of sets is being used in almost every branch of mathematics. In mathematics, a collection of particular things or a group of particular objects is called a set. Sets are used to define the concept of relations and functions. The study of probability, geometry, etc., requires a good knowledge of sets. The sets can be represented in various forms. This article will discuss the basics of sets theory and the representation of sets in detail.

Representation of Sets

In our talks, we often include a collection of different things like names of players in any particular football team, all the thriller novels, etc.

In mathematics also, we come across collections, like, collection of natural numbers, composite numbers, irrational numbers, etc.

Ideally, a set is a well-defined collection of objects. Here, well-defined means it must be absolutely clear which object belongs to the set and which does not.

Below given are a few examples of sets that we use in mathematics.

\(N\)	Set of all natural numbers
\(Z\)	Set of all integers
\(Q\)	Set of all rational numbers
\(R\)	Set of all real numbers
\({Z^ + }\)	Set of positive integers
\({Q^ + }\)	Set of positive rational numbers
\({R^ + }\)	Set of positive real numbers

Elements of a Set

The objects that make up a set are known as its elements or members. The components of a set are written inside curly brackets and separated by commas in most cases. The set’s name is always spelt out in capital letters. For example,

\(S = \left\{ {{\rm{FC}}\,{\rm{Barcelona,}}\,{\rm{Real}}\,{\rm{Madrid,}}\,{\rm{Atletico}}\,{\rm{Madrid,}}\,{\rm{Sevilla}}\,{\rm{FC}}} \right\}\)

Here \(S\) is the set’s name, whose elements or members are FC Barcelona, Real Madrid, Atletico Madrid and Sevilla FC.

The Greek letter epsilon \( \in \) is used for the words that belong to or is an element of Thus, \(x \in S\) will be read as \(x\) belongs to set \(S\) or as \(x\) is the element of a set \(S\).

The symbol \( \notin \) stands for does not belongs to or is not an element of.

The various ways to represent a set are mentioned as follows:

1. Descriptive method
2. Roster or Tabular method
3. Rule or Set-builder method

Representation of Sets Using Description Method

A well-defined description of the elements of a set is created using this method. The description of items is frequently encased in curly brackets. For instance, a set of tennis players with ages between \(19\) years and \(30\) years

\( = \left\{ {{\rm{Tennis}}\,{\rm{players}}\,{\rm{with}}\,{\rm{ages}}\,{\rm{between}}\,19\,{\rm{years}}\,{\rm{and}}\,30\,{\rm{years}}} \right\}\)

A set of integers greater than \(-5\) but less than \( + 5 = \left\{ {{\rm{Integers}}\,{\rm{greater}}\,{\rm{than}} – 5\,{\rm{and}}\,{\rm{smaller}}\,{\rm{than}}\, + 5} \right\}\)

Representation of Sets Using Roster Method

All of the items of a set are listed in roster form, with commas separating them and braces enclosing them. The order in which the items are listed in a roster is irrelevant. In addition, an element is not usually repeated when writing the set in roster form. For example, the set of all even positive integers less than \(9\) is described in the roster form as \(\left\{ {2,4,6,8} \right\}\). Some more examples of representing a set in roster form are as follows.

1. The set of all natural numbers which divide \(\left\{ {1,2,3,6,7,21} \right\}\).
2. The set of odd numbers less than \(20\) is \(\left\{ {1,3,5,7,9,11,13,15,17,19} \right\}\).

Representation of Sets Using Set-Builder Method

In set-builder form, all elements of a set share a single common attribute that no element outside the set possesses. For example, if \(K\) is the set of counting numbers greater than \(12\), then set \(K\) in set-builder form can be written as

\(K = \) {\(x:x\) is a counting number greater than \(12\)} or it can be written
as \(K = \) {\(x\left| x \right.\) is a counting number greater than \(12\)}.

This will be read as \(K\) is a set of elements \(x\) such that \(x\) is a counting number greater than \(12\).

Representation of Sets Using Venn Diagrams

Venn diagrams can be utilised to depict the majority of the relationships between sets. Rectangles and closed curves, primarily circles, make up these diagrams. The universal set is commonly represented by a rectangle, whereas circles represent its subsets.

The constituents of the sets are written in their corresponding circles in Venn diagrams. For example, in the below given Venn diagram:

\(U = \left\{ {1,2,3,4,5,6,7,8,9,10} \right\}\) is the universal set for which \(A = \left\{ {2,4,6,8,10} \right\}\) and \(B = \left\{ {4,6} \right\}\) are subsets

Union of Sets

The union of two sets \(A\) and \(B\) is the set \(C\) which consists of all those elements which are either in \(A\) or in \(B\). In symbols, we write \(A \cup B = x:x \in A\) or \(x \in B\)

The union of two sets can be represented by the Venn diagram, as shown below.

Intersection of Sets

The intersection of two sets \(A\) and \(B\) is the set of all those elements which belongs to both \(A\)and \(B\). Symbolically, we write the intersection as, \(A \cap B = x:x \in A\) or \(x \in B\)

The intersection of two sets can be represented by the Venn diagram, as shown below:

Disjoint Sets

If \(A\) and \(B\) are two sets such that \(A \cap B = \emptyset \), then \(A\) and\(B\) are called disjoint sets. The disjoint sets can be represented using the Venn diagram as shown below:

Solved Examples – Representation of Sets

Q.1. State whether or not the following elements form a set; if not, give a reason.
a) All tasty fruits.
b) All the girls in your class whose height are less than your height.
c) All the easy problems in your textbook.
d) The first \(6\) counting numbers.
Ans: a) No, the fruit may be tasty for one person may not be tasty for the other person. Thus, objects are not well-defined.
b) Yes, it is a set.
c) No, some problems may be easy for one student but may not be easy for some other students. Thus objects are not well-defined.
d) Yes, it is a set.

Q.2. Write the solution set of the equation \({x^2} + x – 2 = 0\) in roster form.
Ans: The given equation can be written as \((x – 1)(x + 2) = 0\) i.e.,\(x = 1,\, – 2\).
Therefore, the solution set of the given equation can be written in the roster form as \(\left\{ {1, – 2} \right\}\).

Q.3. Write the set {\(x:x\) is a positive integer and \({x^2} < 70\)} in the roster form.
Ans: The required numbers are \(1,\,2,\,3,\,4,\,5,\,6,\,7,\,8\). So, the given set in the roster form is \(\left\{ {1,2,3,4,5,6,7,8} \right\}\).

Q.4. If \(X\) and \(Y\) are two sets such that \(X \cup Y\) has \(50\) elements., \(X\) has \(28\) elements, and \(Y\) has \(32\) elements, how many elements did \(X \cap Y\) have?
Ans:  Given that, \(n(X \cup Y) = 50,\,n(X) = 38,\,n(Y) = 22,\,n(X \cap Y) = \)?
We know that, \(n(X \cup Y) = n(X) + n(Y) – n(X \cap Y)\)

Rearranging the formula, we get, \(n(X \cap Y) = n(X) + n(Y) – n(X \cup Y)\)
\( = 38 + 22 – 50 = 10\)

Q.5. Write each of the following sets in the roster form and set-builder form.
a) Set of all three-digit numbers that are perfect square
b) Set of all natural numbers that can divide \(12\) completely.
Ans: a) In the roster form: \(A = \{ 100,\,121,\,144,\,169,\,196,\,225,\,289,\,361,\,400,\,441,\,484,\,529,\,576,\,625,\)
\(729,\,784,\,841,\,900,\,961\} \)
In the set-builder form: \(A = \left\{ {x:x\,{\rm{is}}\,{\rm{a}}\,{\rm{perfect}}\,{\rm{square}}\,{\rm{of}}\,{\rm{three – digit}}\,{\rm{number}}} \right\}\)
b) In the roster form: \(B = \left\{ {1,2,3,4,6,12} \right\}\)
In the set-builder form: \(B = \left\{ {x:x\,{\rm{is}}\,{\rm{a}}\,{\rm{natural}}\,{\rm{number}}\,{\rm{which}}\,{\rm{divides}}\,12\,{\rm{completely}}} \right\}\)

Summary

In this article, we first learnt the basic concept of set and learnt it well with the help of its definition and examples. Then we learnt about the ways of representing the set. We learnt that there are three ways to represent a set. Then we learnt each type in detail with the help of examples, and lastly, we solved some examples to strengthen our grip over the concept.

Frequently Asked Questions (FAQs) – Representation of Sets

Q.1. Define a set with the help of an example.
Ans: A set is a collection of well-defined objects. Well-defined means it must be obvious which objects belong to the set and which does not. A bunch of lovely flowers, for example, is not a set since the flowers in the set are not well-defined. The term “beautiful” is subjective. What is beautiful to one person may not be so to another, so a bunch of lovely flowers cannot be considered a set. On the other hand, if we have a collection of red flowers, it will form a set because each red flower will be included.

Q.2. What are the three ways in representing a set?
Ans: The three ways to represent a set are given as follows:
1. Descriptive method
2. Roster or Tabular method
3. Rule or Set-builder method

Q.3. What does the U represent in sets?
Ans: \(U\) represents a universal set. A universal set is a set that contains all the elements of the other given sets. For example, \(A = \left\{ {1,2,3,4,7} \right\},\,B = \left\{ {4,5,6,7} \right\},\,C = \left\{ {6,7,8,9,10} \right\}\)\), then, we can represent universal set as \(U = \left\{ {1,2,3,4,5,6,7,8,9,10} \right\}\). Therefore, \(A \subset U,\,B \subset U,\,C \subset U\).

Q.4. How do you represent numbers in a set?
Ans: We can represent a number in a set with the help of a roster or tabular method.

Q.5: What are the types of sets?
Ans: The types of sets are:
1. Finite set
2. Infinite set
3. The empty set or the Null set
4. Singelton set
5. Equal set
6. Equivalent sets
7. Disjoint sets
8. Overlapping sets
9. Power set
10. Subset
11. Universal set

Learn About Circumference of Circle

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