Set representation method in Mathematics

Methods of representation of a set

• We have learnt Definition of set, In this article we will know about methods of representation of a set
• Sets are usually denoted with capital letters P, Q, R, X, Y, Z, B, C, D, A etc
• The elements of a set are represented with small letters p, q, r, s, x, t, y, z, a, b, etc
• Object, elements and member of set are same things
• Generally, all elements of a set are represented within a pair of curly braces and separated with commas

EXAMPLE

Example1. Set of all letters in the word “mathematics” can be represented as

P = {m, a, t, h, i, c, s, e}

Example2. Set of vowels in English alphabet can be represented as

Q = {a, e, i, o, u}

Here a, e, i, o, u are the elements of set Q and are being separated as

Comma within a pairs of curly braces {,}

“a” is one of the elements in set Q, so we say that “a belongs to Q” symbol ∈ is called “epsilon” and  used for representing “belongs to”

So, we can write a ∈ Q.

Similarly, e ∈ Q

                a ∈ Q

                 i ∈ Q

If element m is belonging to set P, so we can write “m belongs to P” and represented as “m ∈ P”

If element b is not belonging to set P, so we can write “b does not belong to P” and represented as “b ∉ P”

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There are two methods of representing a set

• Roster form or tabular form
• Set Builder form

Roster form or Tabular form   

In the Roster form of representation of set, a set is represented by listing elements, separated with commas, within a pair of curly braces {,}, Let’s look at the examples given below

Examples

• 1.The set of first five of English Alphabet can be represented as P = {a, b, c, d, e}
• 2. The set of vowels of English Alphabet can be represented as Q = {a, e, i, o, u}
• 3. The set of Odd positive integers less than 15 can be represented as R = {1, 5, 7, 9, 11, 13}
• 4.The set of all prime numbers less than 12 can be represented as S = {2, 3, 5, 7, 11}
• 5.The set of all positive integers which divides 12 can be represented as  Z = {1,2,3,4,6,12}
• 6.The set of first four month of the year can be represented as X = {January, February, March, April}
• 7.The set of days of week can be represented as Y = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
• 8.The set of all-natural numbers less than 7 can be represented in Roster form as First you have to identify all-natural numbers, 1,2,3,4,5, ………..Then all-natural numbers less than 7, are  1,2,3,4,5,6,  Now list these above numbers in a pair of curly braces separated with commas, So, the set of all-natural numbers less than 7 will be M = {1, 2, 3, 4, 5, 6}

Important Note

• In the Roster form of representation of a set, the order in which elements are listed is immaterial (makes no difference), If Set M = {1,2,3,4} and Set N = {4,1,2,3} then Both Set M and Set N are same
• In the Roster form of representation of set, elements generally not be repeated, all the elements are taken as distinct. If Set G = {a, b, c, d} and Set H = {a, a, b, c, d} then Both Set G and Set H, are same and generally set H written in the form of set G

Limitations of representation of set in the Roster form

• It is difficult to list large number of elements in the Roster form of set representation Ex. How to write set of all integers in Roster form of set representation?

In this case we will not comfortable to list all integers in a pair of curly braces and separated with commas. because we don’t know where do we start? And where do we end?

So, we have to write as

Set K = {……… -4, -3, -2, -1, 0, 1, 2, 3, 4, ….}

The dots tell us the list of integers continue indefinitely

• The real problem comes with the roster form of representation is the repeated elements in the set. If the elements are few in numbers, we can easily represent the set-in roster form as given below,

Set L = {1, 2, 2, 3, 4, 5, 6, 7, 7},

If the set contains many elements with repeated elements, it is either uncomfortable or impossible to express the elements in roster form of set representation because the elements do not follow a pattern or have a particular sequence.

Set Builder form

In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

Example: The set of all odd natural numbers less than 12

Set N = {1, 3, 5, 7, 9,11} – Roster form of representation

In above set N, all the element possess common property as all elements are odd and less than 12

In Set Builder form we describe the element of the set N by using a symbol x (any other symbol like the letters y, z, etc. could be used) which is followed by a colon “:” or symbol “|”, After the sign of colon “:” or symbol “|” we write the characteristic property possessed by the elements of the set N and then enclose the whole description within a pair of curly braces

N = {x: characteristic property of element x of set N}

or

N = {x | characteristic property of element x of set N}

The curly braces stand for “the set of all”

The symbol “:” and “|” is read as “such that”

And the set N read as “the set of all x such that x is odd natural number less than 12”

So, above set N can be represented in Set Builder form as

N = {x : x is a natural number less than 12}

Or

N = {x | x is a odd natural number less than 12}

EXERCISE 2.1

Set Theory(Methods of representation of set) Class11 Mock Test 2.1(Exercise 2.1)

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Q1. Which of these sets represent vowels of the English alphabet in roster form?

 

 (A) Z = {x | x is a vowel in English Alphabet}

 (B) Z = {x: x is a vowel in English Alphabet}

 (C) Z = {a, e, i, o, u}

 (D) Z = {vowels of the English Alphabet}

Q2. The set of Natural numbers less than 7 in Roster form, represented as?

 

 (A) X = {x | x ∈ N and x ˂ 7}

 (B) X = {x : x ∈ N and x ˂ 7 }

 (C) X = {1, 2, 3, 4, 5, 6}

 (D) Z = {vowels of the English Alphabet}

Q3. A set P = {x: x is an integer and -2 ≤ x < 4} written in the set builder form, in Roster form we can write set P as?

 

 (A) P = {-2, -1, 0,1, 2, 3, 4}

 (B) P = {-2, -1, 0,1, 2, 3}

 (C) P = {x | x is an integer and -2 ≤ x < 4}

 (D) None of the above

Q4. Let set X = {1,2,3,4,5}, element 3 belonging to set X represented as?

 

 (A) X = {3}

 (B) 3 ∈ X

 (C) 3 ∉ X

 (D) None of the above

Q5. In Q4 element 6 is not belonging to set X represented as?

 

 (A) X = {3}

 (B) 3 ∈ X

 (C) 6 ∉ X

 (D) None of the above

Q6. Write the set Q = {x: x ∈ N, x ≤ 6} in Roster form?

 

 (A) Q = {1, 2, 3, 4, 5}

 (B) Q = {1, 2, 3, 4, 5, 6}

 (C) Q = {x | x is a natural number and x ≤ 6}

 (D) None of the above

Q7. Write the set Z = {x | x = 2ⁿ, n ∈ N and n ≤ 3} in Roster form?

 

 (A) Z = {2, 4, 8}

 (B) Z = {1, 2, 3}

 (C) Z = {1, 4}

 (D) None of the above

Q8. The set E = {1, 4, 9, 16, 25, 36} written in roster form, Write set E in set builder form?

 

 (A) E = {x | x = n2, n ∈ N and n ≤ 6}

 (B) E = {x: x = n2, n ∈ N and n ≤ 6}

 (C) Above both are correct

 (D) E = {x: x = n2, n ∈ N and n ˂ 6}

Q9. Write set of all letters in the word “CALCULAS” in Roster form?

 (A) X = {C, A, L, U, S}

 (B) X = {C, A, L, C, U, L, A, S}

 (C) X = C, A, L, U, S

 (D) X = Both A and C are right

Q10. Write set of all odd natural numbers in Set – builder form?

 

 (A) E = {x | x = n2, n ∈ N}

 (B) E = {x: x = 2n-1, n ∈ N}

 (C) Above both are correct

 (D) E = {x: x = 2n, n ∈ N}

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