Sets

Sets in Mathematics are the collection of definite objects that can form a group. The collection of numbers, days of the week, types of vehicles, etc., are a group of items in a set. Each item in the set is considered an element of the set. While writing a set, it is necessary to use curly brackets. This article will highlight sets and their representation and types and discuss the use of Venn diagrams.

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Sets: Definitions

A collection of well-defined and distinct objects is referred to as sets.

Example: Collection of books written by Tagore – given a book, you can say whether Tagore has written it or not. So, this is a set.

Sets: Representation

A set can be represented in the following methods:

1. Description method
2. Roster method or tabular form
3. Rule method or set builder form
4. Venn diagram

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Description Method

In this method, the well-defined description of the elements of the set is written that is enclosed in curly brackets.

Example:

The set of the natural numbers that are less than the number \(10\) is written as \(A = \{ \)natural numbers less than \(10\} \)
The set of months of the year starting with \(J\) is written as \(B = \{ \)months of a year starting with letter \(J\} \)

Roster Method or Tabular Form

You can list out all the elements of a set inside the curly brackets and separate them by the commas.

Practice Exam Questions

Example:

The set of the whole numbers less than the number \(5\) is written as \(A = \{ 0,1,2,3,4\} .\)
The set of the two-digit numbers whose sum of the digits is the number \(9\) is written in the brackets as \(B = \{ 18,27,36,45,54,63,72,81,90\} \)

Rule Method or the Set Builder Form

In this method, you can write a variable representing any member of the set followed by the property, rule or the statement fulfilled by each of the members of a set and then enclose it in the curly brackets.

Example:

The set of all the natural numbers which are divisible by the number \(5\) and less than \(50\) is written as \(B = \{ x:x = 5n,n \in N\) and \(n < 10\} .\)
The set of the factors of the number \(36\) is written as \(c = \{ x\mid x\,{\rm{is}}\,{\rm{a}}\,{\rm{factor}}\,{\rm{of}}\,36\} .\)

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Venn Diagram

It was developed by the mathematician John Venn. In these diagrams, the universal set \((U)\) is represented by a rectangle, and the sets within are represented by circles.

Example: The given Venn diagram shows set \(A\) within the universal set \(U,\) and \(A’\) is the shaded portion outside the circle known as a complement of \(A.\)

Suppose \(U = \{ 16,17,18,19,20\} \) and \(A = \{ 16,19,20\} .\)This can be represented by the Venn diagram given below. From the Venn diagram, you can observe that, \(A’ = \{ 17,18\} \)

Types of Sets

Finite Set

If the elements of the set can be counted, then the set is known as a finite set.

Example: The set of the natural satellites of Jupiter is a finite set.
The set of the two-digit prime numbers is a finite set.

Infinite Set

If the set elements cannot be counted, then the set is known as an infinite set.

Example: The set of fractions lying between the numbers \(1\) and \(2.\)
The set of integers less than the number \(10\) is an infinite set.

Empty Set

The set that contains no element is known as an empty set. It is denoted by the symbol \(\left\{ {} \right\}\) or \({\phi .}\) The Cardinal number of the empty set is zero, i.e., \(n = \left( \phi \right) = 0.\)

Note: \(\left\{ 0 \right\}\) and \(\left\{ \phi \right\}\) are not empty sets as each of them has one element.

Example: The set of the prime numbers between the numbers \(5\) and \(7\) is an empty set.

Singleton Set

The set that contains only one element is known as a singleton set.

Example: \(A = \{ {\rm{the}}\,{\rm{number}}\,{\rm{of}}\,{\rm{the}}\,{\rm{stars}}\,{\rm{in}}\,{\rm{our}}\,{\rm{solar}}\,{\rm{system}}\} \)  
\(B = \{ {\rm{10}}\} \)

Universal Set

The set that contains all the elements of the sets under consideration is known as the universal set. It differs from set to set and is denoted by \(U.\) We can have more than one universal set for the same set.

Example: If \(A = {\rm{\{ Earth,Mars,Jupiter\} ,}}\) the universal set may be \(\left\{ {{\rm{plantes}}\,{\rm{of}}\,{\rm{the}}\,{\rm{solar}}\,{\rm{system}}} \right\}{\rm{.}}\)
If \(A = \{ 1,3,5,7,9, \ldots \} ,\) then \(U = \{ x:x \in N\} \)

Equal Sets

Two sets \(A\) and \(B\) are considered to be equal if they have the same elements. It is written as \(A=B.\)

Example: Let \(A = \{ 1,3,5,7\} \) and \(B = \{ 3,7,5,1\} ,\) then the elements \(A=B,\) as the elements contained in sets \(A\) and \(B\) are exactly the same.Let \(A = \{ – 3, – 2, – 1,0,1,2,3\} \) and \(B = \left\{ {x:x \in Z,{x^2} \le 9} \right\},\) then \(A = B.\)

Equivalent Sets

Two sets \(A\) and \(B\) are considered to be equivalent sets if they have the same number of elements. The elements may be or may not be identical. Thus, two the sets \(A\) and \(B\) are equivalent if \(nA=n(B),\) and it is represented as \(A↔B\) (you can read as \(A\) is equivalent to \(B\)).

Example: Let \(A = {\rm{\{ colours}}\,{\rm{of}}\,{\rm{the}}\,{\rm{rainbow\} }}\,{\rm{B = \{ x:x}}\,{\rm{is}}\,{\rm{the}}\,{\rm{prime}}\,{\rm{number}}\,{\rm{less}}\,{\rm{than}}\,{\rm{19\} ,}}\) then \(n\left(A \right) = 7 = n\left( B \right)\). So, \(A↔B.\)Let \(A = \{ {\rm{letters}}\,{\rm{of}}\,{\rm{the}}\,{\rm{word}}\,{\rm{FLOWER}}\} ,B = {\rm{\{ letters}}\,{\rm{of}}\,{\rm{the}}\,{\rm{word}}\,{\rm{FOLLOWER\} }}.\) Here, each set is expressed as \(\{ {\rm{F}},{\rm{L}},{\rm{O}},{\rm{W}},{\rm{E}},{\rm{R}}\} \Rightarrow n(A) = 6 = n(B).\) So, \(A↔B.\) As the elements of both the sets are identical, and we say \(A=B.\)

Union of Sets

Union of a couple of sets \(A\) and \(B\) is a set that consists of all the elements of set \(A\) as well as all the elements of set \(B.\) it is denoted by \(A \cup B\) and is read as \(‘{\rm{A}}\) union \(B’.\)

Example: If the sets \(A = \{ 1,2,3\} \) and \(B = \{ 2,3,5,6\} ,\) then \(A \cup B = \{ 1,2,3,5,6\} .\)If \(A = \{ x:x\,{\rm{is}}\,{\rm{divisible}}\,{\rm{by}}\,{\rm{2,}}\,x \in N\} \) and \(B = \{ x:x\,{\rm{is}}\,{\rm{divisible}}\,{\rm{by}}\,{\rm{3,}}\,x \in N\} ,\) then \(A \cup B = \{ x:x\,{\rm{is}}\,{\rm{divisible}}\,{\rm{by}}\,{\rm{either}}\,{\rm{the}}\,{\rm{numbers}}\,{\rm{2}}\,{\rm{or}}\,{\rm{3}}\,{\rm{or}}\,{\rm{both;}}\,x \in N\} .\)

Intersection of Sets

The intersection of two sets \(A\) and \(B\) is a set represented by \({\rm{A}} \cap B,\) that contains elements of both sets \(A\) and \(B.\)

Example: If the sets \(A = \{ 1,2,3\} \) and \(B = \{ 2,3,5,6\} ,\) then \({\rm{A}} \cap B = \{ 3\} .\)

Complement of a Set

The complement of a set \(A,\) represented by \(A’\) contains all the elements of its corresponding universal set that are not contained in set \(A.\)

Example: If the universal set is \(U = \{ 1,2,3,4,5,6,7,8,9,10\} \) and \(A = \{ 2,3,5,6\} ,\) then, the complement of the set \(A,\) denoted by \(A’\) is given by, \({A^\prime } = \{ 1,4,7,8,9,10\} .\)

Difference of Two Sets

If the two sets are \(A\) and \(B,\) then their difference, i.e., \(A-B\) is a set of those elements that belong to \(A\) but not to \(B.\)

Example: Let \(A = \{ 6,7,8,9\} \) and \(B = \{ 4,6,8,10\} ,\) then \(A – B = \{ 7,9\} \) and \(B – A\{ 4,10\} .\)

Symmetric Difference of Two Sets

If the two sets are \(A\) and \(B,\) then the symmetric difference between them is given by \((A – B) \cup (B – A).\)

Example:  Let \((A = \{ 6,7,8,9\} \) and \(B = \{ 4,6,8,10\} ,\) then \(A – B = \{ 7,9\} \) and \(B – A\{ 4,10\} .\)Then, the symmetric difference between \(A\) and \(B\) is \((A – B) \cup (B – A) = \{ 4,7,910\} \)

Sets in Python

The set in Python is an unordered collection the data type that is iterable, mutable and has no duplicate elements. Python’s class of sets contains the mathematical notions. Like you know, the sets are unordered. You cannot access the items using indexes as you do in the lists.

Data Sets

A data set is the collection of the data. The data set lists values for variables, such as the height and weight of an object, for each data set member. Each value is called the datum. Data sets can also consist of the collection of documents or files.

Example: The number of fish eaten by each of the dolphins at the aquarium is a data set.

Solved Examples

Q.1. Write the following sets using the described method.
(a) \(A = \{ x:x = 5n,n \in Z\,{\rm{and}}\, – 5 \le n \le 5\} \)
(b) \(B = \{ 1,8,27,64,125\} \)

Ans: We have
(a) \(A = \{ x:x = 5n,n \in Z\)
\( – 5 \le n \le 5\} \to A = \{ {\rm{multiples}}\,{\rm{of}}\,{\rm{5}}\,{\rm{lying}}\,{\rm{between}} – 25\,{\rm{and}}\,{\rm{25}}\,{\rm{both}}\,{\rm{inclusive\} }}\)
(b) \(B = \{ 1,8,27,64,125\} \to B = {\rm{\{ cubes}}\,{\rm{of}}\,{\rm{first}}\,{\rm{five}}\,{\rm{natural}}\,{\rm{numbers\} }}\)

Q.2. Write the given sets in roster form:
(a) \(A = \{ x:x = 3n,n \in Z, – 4 \le n \le 4\} \)
(b) \(B = \{ x:x \in N,3x – 7 < 11\} \)

Ans: We have,
(a) Given, \(n \in Z\) and \( – 4 \le n \le 4,\) i.e. n can take up values \( – 4, – 3, – 2, – 1,0,1,2,3,\) and \(4.\)
\(\therefore A = \{ – 12, – 9, – 6, – 3,0,3,6,9,12\} \)
(b) Given, \(x \in N\) and \(3x – 7 < 11 \Rightarrow 3x < 18 \Rightarrow x < 6\)
So, \(x\) can take up values \(1, 2, 3, 4,\) and \(5,\) since \(x \in N\)
\(\therefore B = \{ 1,2,3,4,5\} \)

Q.3. If \(n(A – B) = 12,n(B – A) = 16\) and \(n(A \cap B) = 5,\) find \(n(A),n(B),n(A \cup B).\)

Ans: We know that, \(n(A – B) = n(A) – n(A \cap B)\)
\(\therefore 12 = n(A) – 5 \Rightarrow n(A) = 17\)
Second, we have that, \(n(B – A) = n(B) – n(A \cap B)\)
\(\therefore 16 = n(B) – 5 \Rightarrow n(B) = 21\)
Again, \(n(B – A) = n(A \cup B) – n(A)\)
\(\therefore 16 = n(A \cup B) – 17 \Rightarrow n(A \cup B) = 33\)

Q.4. In a class of \(40\) pupils, \(32\) like English, \(25\) like French. How many students like both English and French?

Ans: Let \(E = \{ {\rm{pupils}}\,{\rm{who}}\,{\rm{like}}\,{\rm{English\} }}\) and \(F = \{ {\rm{pupils}}\,{\rm{who}}\,{\rm{like}}\,{\rm{French\} }}\)
\(\therefore n(E) = 32,n(F) = 25\) and \(n(E \cup F) = 40\)
Let \(x\) be the number of pupils who like both English and French. The figure shown below represents the given information.

Thus, the number of pupils who like English only \(=32-x\) and the number of pupils who like French only\(=25-x.\)
Total number of pupils\(=40\)
\(\therefore 32 – x + x + 25 – x = 40 \Rightarrow – x = 40 – 57 \Rightarrow x = 17\)
\(\therefore 17\) pupils like both the subjects.
Note: Above given problem can also be solved by using the relation:
\(n(E \cup F) = n(E) + n(F) – n(E \cap F) \Rightarrow 40 = 32 + 25 – n(E \cap F)\)
\( \Rightarrow n(E \cap F) = 57 – 40 = 17\)

Q.5. In a class of \(40\) boys, \(24\) play cricket and \(20\) play volleyball, while \(11\) play nether. Identify the number of boys who play both volleyball and cricket.

Ans: Let \(U = \left\{ {{\rm{boys}}} \right\},C = \left\{ {{\rm{boys}}\,{\rm{who}}\,{\rm{play}}\,{\rm{cricket}}} \right\}\) and
\(V = {\rm{\{ boys}}\,{\rm{who}}\,{\rm{play}}\,{\rm{volleyball\} }}\)
Let \(x\) be the number of boys who play both volleyball and cricket.
So, the number of boys who play only cricket\(=24-x\)
Number of boys who play only volleyball\(=20-x\)
\(n(U)=\) number of boys who play only cricket \(+\) number of boys who play only volleyball \(+\) number of boys who play both \(+\) number of boys who play neither cricket nor volleyball.
\( \Rightarrow 24 – x + 20 – x + x + 11 = 40\)
\( \Rightarrow 55 – x = 40\)
\( \Rightarrow x = 55 – 40\)
\(\therefore x = 15\)
\(\therefore \) Fifteen boys play both cricket and volleyball.

Summary

In the given article, you have learnt about the term set with an example, followed by the representation of sets and then types of sets. Then, we have discussed equivalent sets and the union of sets. We also glanced at the symmetric difference between the two sets. Finally, talked about sets in python and data set and gave some solved examples and a few FAQs.

FAQs on Sets

Frequently asked questions related to sets is listed as follows:

Q. What are examples of the set?
Ans: The set is the well-defined collection of the elements or numbers or objects that are represented within the curly brackets \(\left\{ {} \right\}.\) For example – \(\left\{ {1,2,3,4} \right\}\) is a set of the numbers.

Q. What is \(A \cap B\)?
Answer: The intersection of the two sets \(A\) and \(B,\) denoted by \(A \cap B,\) contains those elements that are present in both sets \(A\) and \(B.\)

Q. How do you define a set?
Ans: A collection of objects that are well defined and distinct is known as a set.
The word well defined signify that given an object, you must be able to find out whether it belongs to a given set or not. The word distinct means that an entity should not be repeated.

Q. Which command sets the number for all lines?
Ans: Vim/Vi is the text editor of choice for many software developers and Linux system administrators. In this system, the absolute line numbering is the standard line numbering, which displays the appropriate line number next to each line of text. To activate the line numbering, a set of the number flags are activated by pressing the “Esc” key to switch to command mode. The command that triggers this set of the number flag for all the lines for this activity is \({\rm{set}}\,{\rm{nu}}{\rm{.}}\)

Q. What are the types of sets?
Ans: The type of sets are given below:
1. Finite set
2. Infinite set
3. Empty set
4. Singleton set
5. Universal set
6. Equal set
7. Equivalent set
8. Overlapping set
9. Disjoint set

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