Representation of Sets

Sets are generally represented by following two ways

1. Roster or Tabular Form or Listing Method

2. Set—builder Form or Rule Method

1. Roster or Tabular Form or Listing Method

In roster form,all the elements of a set are listed,the elements are being separated by commas and are enclosed within braces {}. For example,the set of all even positive integers less than 7 is described in roster form as {2,4,6}. Some more examples of representing a set in roster form are given below:

(a) The set of all natural numbers which divide 42 is {1,2,3,6,7,14,21,42}.

[Note] In roster form,the order in which the elements are listed is immaterial.
Thus,the above set can also be represented as {1,3,7,21,2,6,14,42}.

(b) The set of all vowels in the English alphabet is {a,e,i,o,u}.

(c) The set of odd natural numbers is represented by {

1,3,5,…}. The dots tell us that the list of odd numbers continue indefinitely.

[Note] It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct. For example,the set of letters forming the word ‘SCHOOL’ is {S,C,H,O,L} or {H,O,L,C,S}. Here,the order of listin; elements has no relevance.

In roster form,all the elements of a set are listed, the elements are being separated by commas and are enclosed within curly braces { }.

e.g.,

(i) The set of all natural numbers less than 10 is represented in roster form as {1,2,3,4,5,6,7,8,9}.

(ii) The set of prime natural numbers is {2,3,5,7,…}. Here,three dots tell us that the list of prime natural numbers continue indefinitely.

[Note]
(i) In roster form,order in which the elements are listed is not important. Hence,the set of natural numbers less than 10 can also be written as {2,4,1,3,5,6,8,7,9} instead of{1,2,3,4,5,6,7,8,9}. Here, the order of listing elements is not important.
(ii) In roster form, element is not repeated i.e., all the elements are taken as distinct. Hence, the set of letters forming the ‘MISCELLANEOUS’ is {M,I,S,C,E,L,A,N,O,U}.

Example 1: Roster Form of Sets
Express the following in roster form.
a) Set G is the set of natural numbers less than 6.
b) Set H is the set of natural numbers less than or equal to 80.
c) Set J is the set of planets in Earth’s solar system.
Solution:
a) The natural numbers less than 6 are 1, 2, 3, 4, and 5. Thus. set G in roster fonn is G={1,2,3,4,5}.
b) H={1,2,3,4, …, 80}. The 80 after the ellipsis indicates that the elements continue in the same manner up to and including the number 80.
c) J={Mercury,Venus,Earth,Mars,Jupiter,Saturn. Uranus,Neptune}

Ex2. Write the following sets in roster form:
(i) A={x:x is an integer and -3B={x:x is a natural number less than 6}.
(iii) C={x:x is a two-digit natural number such that the sum of its digits is 8}
(iv) D={x:x is a prime number which is divisor of 60}.
(v) E=The set of all letters in the word TRIGONOMETRY.
(vi) F=The set of all letters in the word BETTER.
Solution:
(i) A={x:x is an integer and -3A={-2,-1,0,1,2,3,4,5,6}
(ii) B={x:x is a natural number less than 6}
The elements of this set are 1,2,3,4,and 5 only.
Therefore,the given set can be written in roster form as B={1,2,3,4,5}
(iii) C={x:x is a two-digit natural number such that the sum of its digits is 8}. The elements of this set are 17,26,35,44,53,62,71,and 80 only.
Therefore,this set can be written in roster form as
C={17,26,35,44,53,62,71,80}
(iv) D={x:x is a prime number which is a divisor of 60}.

60=2×2×3×5
The elements of this set are 2,3,and 5 only.
Therefore,this set can be written in roster form as D={2,3,5}.
(v) E=The set of all letters in the word TRIGONOMETRY.
There are 12 letters in the word TRIGONOMETRY,out of which letters T, R, and O are repeated.
Therefore,this set can be written in roster form as E={T,R,I,G,O,N,M,E,Y}
(vi) F=The set of all letters in the word BETTER
There are 6 letters in the word BETTER,out of which letters E and T are repeated.
Therefore,this set can be written in roster form as F={B,E,T,R}.

2. •Set—builder Form or Rule Method