Understanding Sets

What is a set?

A set is a collection of well-defined and distinct objects. Every object of the collection forming a set is called a member or element of the set. When an object is a member of a set we say that the object belongs to the set.

Any collection of objects is not a set. Let us consider the following collection of objects.

(i) The first four natural numbers

(ii) The collection of the letters of the word TERRACOTTA

(iii) Four consecutive whole numbers

The members of collection (i) are 1, 2, 3 and 4. These objects are well-defined. Now we have to decide whether a particular object belongs to the collection or not. For example, 3 is a member of the collection, while 5 is no. Also, the members of the collection are distinct or different from each other. Thus, the collection is a set.

But collection (ii) is not a set because letter ‘R’, ‘A’ and ‘T’ occurs in it twice, and hence, the objects belonging to the collection are not distinct. If we take only one ‘R’, only one ‘A’ and only one ‘T’ in place of two ‘R’s, ‘A’s and ‘T’s, we get the collection T, E, R, A, C, O which is a set.

Again, collection (iii) is not a set as we are unable to decide whether an object belongs to the collection or not. Here, in this case 3 may belong to the collection or it may not. The same is true for any other whole number.

Notation:

Usually, sets are denoted by capital letters, such as A, B, C, X and Y. the members or elements of a set are denoted by small letters, such as a, b, c, x and y. If ‘a’ is a member of the set A, we write it as “”, which is read as “a belongs to A”. If ‘a’ is not a member of the set A, we write it as “”, which is read as “a does not belong to A”.

Example: Let A be the set of the first four odd natural numbers. Then, .

Notations for some special sets:

(i) The set of natural numbers is denoted by . As 8 is a natural number, . But is not a natural number. So, .

(ii) The set of whole numbers is denoted by . As 0 is a whole number, . But is not a whole number. So, .

(iii) The set of integers is denoted by or . As is an integer, . But is not an integer. So, .

Representation of a set:

A set can be represented in two ways:

(i) Roster method or Tabular Form

(ii) Rule Method or Set-builder Form

Roster Method or Tabular Form:

In this method, the members of a set are listed inside the braces {} and separated from each other by commas. If an object appears more than once in a collection, we write it only once.

Examples: (i) The set A of the first four natural numbers is written as A={1,2,3,4}.

(ii) The set X of the letters of the word KOLKATA is written as X={K, O, L, A, T}

Note: The members of a set can be listed in any order. Thus, the set {1, 2, 3, 4} can be written as {2, 3, 4, 1}.

Rule Method or Set-builder Form:

If the members of a set have a common property then they can be determined by describing the property. For example, the members of the set A={1. 2, 3, 4, 5} have a common property, namely, all are natural numbers less than 6. No other natural number has this property. So, we can write the set A as follows.

A ={   is a natural number less than 6} which is read as “A is the set of members such that is a natural number less than 6”.

The above set can also be written more precisely as A ={ }.

Sometimes the set A is also written as A = {the set of all natural numbers less than 6}.

In this case, the description of the common property of the members is given inside the braces {}. This is a crude form of the rule method.

A set given in the set-builder form can be expressed in the tabular form by listing the objects satisfying the rule.

Examples:

(i) X={ is a letter of the word WOOD}

is any one of the letters, W, O, D. So, X= {W, O, D}.

(ii) A= { is a whole number greater than 6 but less than 10}

is any one of the numbers 7, 8 and 9. So, A= {7, 8, 9}.

A set given in the tabular form can be expressed in the set-builder form if the elements obey some common property. But the representation may not be unique.

Examples: Y= {0, 2, 4}.

Here, each of the elements is an even whole number that is less than 5. No other whole number has this property.

So, Y= { is an even whole number less than 5}.

Example 1: Which of the following collections are sets? Give reasons.

(a) All even whole numbers less than 9.

(b) All the beautiful girls of your school.

Solution:

(a) This is a set because it is possible to decide whether or not a number is a member of the collection.

(b) This is not a set, as there is no definite way to decide whether a particular girl is beautiful or not.

Example 2: Let X be the set of the natural numbers which are multiples of 3 and are less than 12. Fill in the blanks using or .

(a) 6 ___ X

(b) 10 ___ X

(c) 12 ___ X

Solution:

(a) [ 6 is a multiple of 3 and it is less than 12.]

(b) [ 10 is not a multiple of 3.]

(c) [ 12 is not less than 12 though it is a multiple of 3.]

Example 3: (a) Write the set X of odd natural numbers lying between 4 and 10 in the tabular as well as the set-builder form.

Solution:

In the tabular form: X= {5, 7, 9}

In the set-builder form: X= { }

(b) A={ }. Write A using the roster method.

Solution:

and , we get .

.

A={6, 8, 10, 12, 14}.

Exercise:

1. Which of the following collections are sets?
   1. All the months of a year.
   2. All the letters of the word MIZORAM.
   3. All the natural numbers less than 20 which are perfect squares.
   4. All super actors of India.
   5. All the integers lying between -5 and 5.
   6. Four consecutive integers.
   7. All prime numbers lying between 4 and 12.
   8. All the good boys of our neighbourhood.
2. Write the set A= { and } by the roster method and fill in the blanks by using or :
   1. 11 ____ A
   2. 5 ____ A
   3. 17 ____ A
   4. 10 ____ A
3. Write the following sets by the roster as well as the rule method:
   1. The set of letters of the word EXAMINATION.
   2. The set of consonants in the word RABINDRANATH.
   3. The set of even integers lying between -10 and 10.
   4. The set of whole numbers lying between -4 and 2.
   5. The set of natural numbers less than 11.
4. Write the following sets in the tabular form.
   1. A= { }
   2. B= { is a prime number of one digit}
   3. C= { is a letter of the word COMPUTER}
   4. D= { is a two-digit number divisible by 15}
5. Write the following sets in the set-builder form:
   1. {a, e, i, o, u}
   2. {0, 5, 10, 15}
   3. {1, 3, 5, 7, ……..}