Sets

An introduction of sets and its definition in mathematics. The concept of sets is used for the foundation of various topics in mathematics.

To learn sets we often talk about the collection of objects, such as a set of vowels, set of negative numbers, a group of friends, a list of fruits, a bunch of keys, etc.

What is set (in mathematics)?

The collection of well-defined distinct objects is known as a set. The word well-defined refers to a specific property which makes it easy to identify whether the given object belongs to the set or not. The word ‘distinct’ means that the objects of a set must be all different.

For example:

1. The collection of children in class VII whose weight exceeds 35 kg represents a set.

2. The collection of all the intelligent children in class VII does not represent a set because the word intelligent is vague. What may appear intelligent to one person may not appear the same to another person.

Elements of Set:

The different objects that form a set are called the elements of a set. The elements of the set are written in any order and are not repeated. Elements are denoted by small letters.

Notation of a Set:

A set is usually denoted by capital letters and elements are denoted by small letters

If x is an element of set A, then we say x ϵ A. [x belongs to A]

If x is not an element of set A, then we say x ∉ A. [x does not belong to A]

For example:

The collection of vowels in the English alphabet.

Solution :

Let us denote the set by V, then the elements of the set are a, e, i, o, u or we can say, V = [a, e, i, o, u].

We say a ∈ V, e ∈ V, i ∈ V, o ∈ V and u ∈ V.

Also, we can say b ∉ V, c ∉ v, d ∉ v, etc.

How to state that whether the objects form a set or not?

1. A collection of ‘lovely flowers’ is not a set, because the objects (flowers) to be included are not well-defined.

Reason: The word “lovely” is a relative term. What may appear lovely to one person may not be so to the other person.

2. A collection of “Yellow flowers” is a set, because every red flowers will be included in this set i.e., the objects of the set are well-defined.

3. A group of “Young singers” is not a set, as the range of the ages of young singers is not given and so it can’t be decided that which singer is to be considered young i.e., the objects are not well-defined.

4. A group of “Players with ages between 18 years and 25 years” is a set, because the range of ages of the player is given and so it can easily be decided that which player is to be included and which is to be excluded. Hence, the objects are well-defined.

Now we will learn to state which of the following collections are set.

State, giving reason, whether the following objects form a set or not:

(i) All problems of this book, which are difficult to solve.

Solution:

The given objects do not form a set.

Reason: Some problems may be difficult for one person but may not be difficult for some other persons, that is, the given objects are not well-defined.

Hence, they do not form a set.

(ii) All problems of this book, which are difficult to solve for Aaron.

Solution:

The given objects form a set.

Reason: It can easily be found that which are difficult to solve for Aaron and which are not difficult to solve for him.

Hence, the objects form a set.

(iii) All the objects heavier than 28 kg.

Solution:

The given objects form a set.

Reason: Every object can be compared, in weight, with 28 kg. Then it is very easy to select objects which are heavier than 28 kg i.e., the objects are well-defined.

Hence, the objects form a set.

The members (objects) of each of the following collections form a set:

(i) students in a class-room

(ii) books in your school-bag

(iii) counting numbers between 5 to 15

(iv) students of your class, which are taller than you and so on.

What are the elements of a set or members of a set?

The objects used to form a set are called its element or its members.

Generally, the elements of a set are written inside a pair of curly (idle) braces and are represented by commas. The name of the set is always written in capital letter.

Solved Examples to find the elements or members of a set:

1. A = {v, w, x, y, z}

Here ‘A’ is the name of the set whose elements (members) are v, w, x, y, z.

2. If a set A = {3, 6, 9, 10, 13, 18}. State whether the following statements are ‘true’ or ‘false’:

(i) 7 ∈ A

(ii) 12 ∉ A

(iii) 13 ∈ A

(iv) 9, 12 ∈ A

(v) 12, 14, 15 ∈ A

Solution:

(i) 7 ∈ A

False, since the element 7 does not belongs to the given set A.

(ii) 10 ∉ A

False, since the element 10 belongs to the given set A.

(iii) 13 ∈ A

True, since the element 13 belongs to the given set A.

(iv) 9, 10 ∈ A

True, since the elements 9 and 12 both belong to the given set A.

(v) 10, 13, 14 ∈ A

False, since the element 14 does not belongs to the given set A.

3. If set Z = {4, 6, 8, 10, 12, 14}. State which of the following statements are ‘correct’ and which are ‘wrong’ along with the correct explanations

(i) 5 ∈ Z

(ii) 12 ∈ Z

(iii) 14 ∈ Z

(iv) 9 ∈ Z

(v) Z is a set of even numbers between 2 and 16.

(vi) 4, 6 and 10 are members of the set Z.

Solution:

(i) 5 ∈ Z

Wrong, since 5 does not belongs to the given set Z i.e. 5 ∉ Z

(ii) 12 ∈ Z

Correct, since 12 belongs to the given set Z.

(iii) 14 ∈ Z

Correct, since 14 belongs to the given set Z.

(iv) 9 ∈ Z

Wrong, since 9 does not belongs to the given set Z i.e. 9 ∉ Z

(v) Z is a set of even numbers between 2 and 16.

Correct, since the elements of the set Z consists of all the multiples of 2 between 2 and 16.

(vi) 4, 6 and 10 are members of the set Z.

Correct, since the 4, 6 and 10 those numbers belongs to the given set Z.

What are the different types of sets?

The different types of sets are explained below with examples.

Empty Set or Null Set:

A set which does not contain any element is called an empty set, or the null set or the void set and it is denoted by ∅ and is read as phi. In roster form, ∅ is denoted by {}. An empty set is a finite set, since the number of elements in an empty set is finite, i.e., 0.

For example: (a) The set of whole numbers less than 0.

(b) Clearly there is no whole number less than 0.

Therefore, it is an empty set.

(c) N = {x : x ∈ N, 3 < x < 4}

 Let A = {x : 2 < x < 3, x is a natural number}

Here A is an empty set because there is no natural number between
2 and 3.

 Let B = {x : x is a composite number less than 4}.

Here B is an empty set because there is no composite number less than 4.

Note:

∅ ≠ {0} ∴ has no element.

{0} is a set which has one element 0.

The cardinal number of an empty set, i.e., n(∅) = 0

Singleton Set:

A set which contains only one element is called a singleton set.

For example:

 A = {x : x is neither prime nor composite}

It is a singleton set containing one element, i.e., 1.

 B = {x : x is a whole number, x < 1}

This set contains only one element 0 and is a singleton set.

 Let A = {x : x ∈ N and x² = 4}

Here A is a singleton set because there is only one element 2 whose square is 4.

 Let B = {x : x is a even prime number}

Here B is a singleton set because there is only one prime number which is even, i.e., 2.

Finite Set:

A set which contains a definite number of elements is called a finite set. Empty set is also called a finite set.

For example:

 The set of all colors in the rainbow.

 N = {x : x ∈ N, x < 7}

 P = {2, 3, 5, 7, 11, 13, 17, …… 97}

Infinite Set:

The set whose elements cannot be listed, i.e., set containing never-ending elements is called an infinite set.

For example:

 Set of all points in a plane

 A = {x : x ∈ N, x > 1}

 Set of all prime numbers

 B = {x : x ∈ W, x = 2n}

Note:

All infinite sets cannot be expressed in roster form.

For example:

The set of real numbers since the elements of this set do not follow any particular pattern.

Cardinal Number of a Set:

The number of distinct elements in a given set A is called the cardinal number of A. It is denoted by n(A).

For example:

 A {x : x ∈ N, x < 5}

A = {1, 2, 3, 4}

Therefore, n(A) = 4

 B = set of letters in the word ALGEBRA

B = {A, L, G, E, B, R}

Therefore, n(B) = 6

Equivalent Sets:

Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B). The symbol for denoting an equivalent set is ‘↔’.

For example:

A = {1, 2, 3} Here n(A) = 3

B = {p, q, r} Here n(B) = 3

Therefore, A ↔ B

Equal sets:

Two sets A and B are said to be equal if they contain the same elements. Every element of A is an element of B and every element of B is an element of A.

For example:

A = {p, q, r, s}

B = {p, s, r, q}

Therefore, A = B

The various types of sets and their definitions are explained above with the help of examples.