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NCERT Solutions · Class 11 Mathematics Sets

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EXERCISE 1.1 1–6 (part 1 of 6)

  1. Exercise 1

    Which of the following are sets? Justify your answer.
    (i)
    The collection of all the months of a year beginning with the letter J.
    (ii)
    The collection of ten most talented writers of India.
    (iii)
    A team of eleven best-cricket batsmen of the world.
    (iv)
    The collection of all boys in your class.
    (v)
    The collection of all natural numbers less than 100.
    (vi)
    A collection of novels written by the writer Munshi Prem Chand.
    (vii)
    The collection of all even integers.
    (viii)
    The collection of questions in this Chapter.
    (ix)
    A collection of most dangerous animals of the world.

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    NCERT’s answer
    (i)
    , (iv), (v), (vi), (vii) and (viii) are sets.
    Well-defined test. A collection is a set only when, for every object, it can be decided beyond dispute whether the object belongs to it. Words such as best, most talented and most dangerous record an opinion, so collections described by them fail the test.
    (i)
    The months are January, June and July — fixed by their names. Set.
    (ii)
    "Ten most talented writers" varies from person to person. Not a set.
    (iii)
    "Eleven best batsmen" varies from person to person. Not a set.
    (iv)
    The boys in your class can be named exactly. Set.
    (v)
    For any number we can check whether it is a natural number below \(\displaystyle 100\). Set.
    (vi)
    The novels written by Munshi Prem Chand form a definite, recorded list. Set.
    (vii)
    An integer either is even or is not; there is no ambiguity. Set.
    (viii)
    The questions printed in this chapter are fixed. Set.
    (ix)
    "Most dangerous animals" is a matter of opinion. Not a set.
    Sets: (i), (iv), (v), (vi), (vii), (viii). Not sets: (ii), (iii), (ix).

  2. Exercise 2

    Let A={1,2,3,4,5,6}\displaystyle \mathrm{A}=\{1,2,3,4,5,6\}. Insert the appropriate symbol ∈\displaystyle \in or ∉\displaystyle \notin in the blank spaces:
    (i)
    5...A
    (ii)
    8\displaystyle 8 ... A
    (iii)
    0...A
    (iv)
    4...A
    (v)
    2. . .A
    (vi)
    10...A

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    Membership test. \(\displaystyle x \in \mathrm{A}\) means \(\displaystyle x\) is one of the elements listed in \(\displaystyle \mathrm{A}\); \(\displaystyle x \notin \mathrm{A}\) means it is not. Here \(\displaystyle \mathrm{A}=\{1,2,3,4,5,6\}\).
    (i)
    \(\displaystyle 5\) appears in the list, so \(\displaystyle 5 \in \mathrm{A}\).
    (ii)
    \(\displaystyle 8\) does not appear, so \(\displaystyle 8 \notin \mathrm{A}\).
    (iii)
    \(\displaystyle 0\) does not appear (the list starts at \(\displaystyle 1\)), so \(\displaystyle 0 \notin \mathrm{A}\).
    (iv)
    \(\displaystyle 4\) appears in the list, so \(\displaystyle 4 \in \mathrm{A}\).
    (v)
    \(\displaystyle 2\) appears in the list, so \(\displaystyle 2 \in \mathrm{A}\).
    (vi)
    \(\displaystyle 10\) does not appear, so \(\displaystyle 10 \notin \mathrm{A}\).
    (i) ∈ (ii) ∉ (iii) ∉ (iv) ∈ (v) ∈ (vi) ∉

  3. Exercise 3

    Write the following sets in roster form:
    (i)
    A={x:x\displaystyle \mathrm{A}=\{x: x is an integer and −3≤x<7}\displaystyle -3 \leq x<7\}
    (ii)
    B={x:x\displaystyle \mathrm{B}=\{x: x is a natural number less than 6\displaystyle 6}\displaystyle \}
    (iii)
    C={x:x\displaystyle \mathrm{C}=\{x: x is a two-digit natural number such that the sum of its digits is 8\displaystyle 8}\displaystyle \}
    (iv)
    D={x:x\displaystyle \mathrm{D}=\{x: x is a prime number which is divisor of 60\displaystyle 60\}
    (v)
    E=\displaystyle \mathrm{E}= The set of all letters in the word TRIGONOMETRY
    (vi)
    F=\displaystyle \mathrm{F}= The set of all letters in the word BETTER

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    NCERT’s answer
    (i)
    $\displaystyle \mathrm{A}=\{-3,-2,-1,0,1,2,3,4,5,6\}$ (ii) $\displaystyle \mathrm{B}=\{1,2,3,4,5\}$ (iii) $\displaystyle \mathrm{C}=\{17,26,35,44,53,62,71,80\}$ (iv) $\displaystyle \mathrm{D}=\{2,3,5\}$ (v) $\displaystyle \mathrm{E}=\{\mathrm{T}, \mathrm{R}, \mathrm{I}, \mathrm{G}, \mathrm{O}, \mathrm{N}, \mathrm{M}, \mathrm{E}, \mathrm{Y}\} \quad$ (vi) $\displaystyle \mathrm{F}=\{\mathrm{B}, \mathrm{E}, \mathrm{T}, \mathrm{R}\}$
    Roster form. List every element once, separated by commas, inside braces. A repeated letter is written only once, because a set records which objects belong, not how many times they occur.
    (i)
    The integers from \(\displaystyle -3\) up to \(\displaystyle 6\) (since \(\displaystyle x<7\)):
    \[\mathrm{A}=\{-3,\,-2,\,-1,\,0,\,1,\,2,\,3,\,4,\,5,\,6\}\]
    (ii)
    The natural numbers below \(\displaystyle 6\) are \(\displaystyle 1,2,3,4,5\):
    \[\mathrm{B}=\{1,2,3,4,5\}\]
    (iii)
    A two-digit number \(\displaystyle \overline{ab}\) needs \(\displaystyle a+b=8\) with \(\displaystyle a \geq 1\). Taking \(\displaystyle a=1,2,\ldots,8\) gives \(\displaystyle b=7,6,5,4,3,2,1,0\):
    \[\mathrm{C}=\{17,26,35,44,53,62,71,80\}\]
    (iv)
    Factorise: \(\displaystyle 60 = 2^{2} \times 3 \times 5\), so the prime divisors are \(\displaystyle 2, 3, 5\):
    \[\mathrm{D}=\{2,3,5\}\]
    (v)
    TRIGONOMETRY has letters T, R, I, G, O, N, O, M, E, T, R, Y; keeping each distinct letter once:
    \[\mathrm{E}=\{\mathrm{T},\mathrm{R},\mathrm{I},\mathrm{G},\mathrm{O},\mathrm{N},\mathrm{M},\mathrm{E},\mathrm{Y}\}\]
    (vi)
    BETTER has letters B, E, T, T, E, R; keeping each distinct letter once:
    \[\mathrm{F}=\{\mathrm{B},\mathrm{E},\mathrm{T},\mathrm{R}\}\]
    A = {−$\displaystyle 3$, −$\displaystyle 2$, −$\displaystyle 1$, $\displaystyle 0$, $\displaystyle 1$, $\displaystyle 2$, $\displaystyle 3$, $\displaystyle 4$, $\displaystyle 5$, $\displaystyle 6$}; B = {$\displaystyle 1$, $\displaystyle 2$, $\displaystyle 3$, $\displaystyle 4$, $\displaystyle 5$}; C = {$\displaystyle 17$, $\displaystyle 26$, $\displaystyle 35$, $\displaystyle 44$, $\displaystyle 53$, $\displaystyle 62$, $\displaystyle 71$, $\displaystyle 80$}; D = {$\displaystyle 2$, $\displaystyle 3$, $\displaystyle 5$}; E = {T, R, I, G, O, N, M, E, Y}; F = {B, E, T, R}

  4. Exercise 4

    Write the following sets in the set-builder form :
    (i)
    (3\displaystyle 3, 6\displaystyle 6, 9\displaystyle 9, 12\displaystyle 12\}
    (ii)
    {2,4,8,16,32}\displaystyle \{2,4,8,16,32\}
    (iii)
    {5,25,125,625}\displaystyle \{5,25,125,625\}
    (iv)
    {2,4,6,…}\displaystyle \{2,4,6, \ldots\}
    (v)
    {1,4,9,…,100}\displaystyle \{1,4,9, \ldots, 100\}

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    NCERT’s answer
    (i)
    $\displaystyle \{x: x=3 n, n \in \mathrm{~N}$ and $\displaystyle 1 \leq n \leq 4\}$ (ii) $\displaystyle \left\{x: x=2^{n}, n \in \mathrm{~N}\right.$ and $\displaystyle \left.1 \leq n \leq 5\right\}$ (iii) $\displaystyle \left\{x: x=5^{n}, n \in \mathrm{~N}\right.$ and $\displaystyle \left.1 \leq n \leq 4\right\}$ (iv) $\displaystyle \{x: x$ is an even natural number $\displaystyle \}$ (v) $\displaystyle \left\{x: x=n^{2}, n \in \mathrm{~N}\right.$ and $\displaystyle \left.1 \leq n \leq 10\right\}$
    Set-builder form. Spot the rule that generates the listed numbers, then write \(\displaystyle \{x : x \text{ satisfies the rule}\}\). (A set can be described in more than one correct way; any equivalent rule is acceptable.)
    (i)
    \(\displaystyle 3,6,9,12\) are the first four multiples of \(\displaystyle 3\):
    \[\{x : x=3n,\; n \in \mathbf{N},\; 1 \leq n \leq 4\}\]
    (ii)
    \(\displaystyle 2,4,8,16,32\) are \(\displaystyle 2^{1},2^{2},2^{3},2^{4},2^{5}\):
    \[\{x : x=2^{n},\; n \in \mathbf{N},\; 1 \leq n \leq 5\}\]
    (iii)
    \(\displaystyle 5,25,125,625\) are \(\displaystyle 5^{1},5^{2},5^{3},5^{4}\):
    \[\{x : x=5^{n},\; n \in \mathbf{N},\; 1 \leq n \leq 4\}\]
    (iv)
    The dots show the list never ends; these are all the even natural numbers:
    \[\{x : x \text{ is an even natural number}\}\]
    (v)
    \(\displaystyle 1,4,9,\ldots,100\) are \(\displaystyle 1^{2},2^{2},3^{2},\ldots,10^{2}\):
    \[\{x : x=n^{2},\; n \in \mathbf{N},\; 1 \leq n \leq 10\}\]
    (i) {x : x = 3n, n ∈ N, $\displaystyle 1$ ≤ n ≤ $\displaystyle 4$} (ii) {x : x = $\displaystyle 2$ⁿ, n ∈ N, $\displaystyle 1$ ≤ n ≤ $\displaystyle 5$} (iii) {x : x = $\displaystyle 5$ⁿ, n ∈ N, $\displaystyle 1$ ≤ n ≤ $\displaystyle 4$} (iv) {x : x is an even natural number} (v) {x : x = n², n ∈ N, $\displaystyle 1$ ≤ n ≤ $\displaystyle 10$}

  5. Exercise 5

    List all the elements of the following sets :
    (i)
    A={x:x\displaystyle \mathrm{A}=\{x: x is an odd natural number }\displaystyle \}
    (ii)
    B={x:x\displaystyle \mathrm{B}=\left\{x: x\right. is an integer, −12<x<92}\displaystyle \left.-\frac{1}{2}<x<\frac{9}{2}\right\}
    (iii)
    C={x:x\displaystyle \mathrm{C}=\left\{x: x\right. is an integer, x2≤4}\displaystyle \left.x^{2} \leq 4\right\}
    (iv)
    D={x:x\displaystyle \mathrm{D}=\{x: x is a letter in the word "LOYAL" }\displaystyle \}
    (v)
    E={x:x\displaystyle \mathrm{E}=\{x: x is a month of a year not having 31\displaystyle 31 days }\displaystyle \}
    (vi)
    F={x:x\displaystyle \mathrm{F}=\{x: x is a consonant in the English alphabet which precedes k}\displaystyle k\}.

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    NCERT’s answer
    (i)
    $\displaystyle \mathrm{A}=\{1,3,5, \ldots\}$ (ii) $\displaystyle \mathrm{B}=\{0,1,2,3,4\}$ (iii) $\displaystyle \mathrm{C}=\{-2,-1,0,1,2\}$ (iv) $\displaystyle \mathrm{D}=\{\mathrm{L}, \mathrm{O}, \mathrm{Y}, \mathrm{A}\}$ (v) $\displaystyle \mathrm{E}=\{$ February, April, June, September, November \} (vi) $\displaystyle \mathrm{F}=\{b, c, d, f, g, h, j\}$
    Roster form. Read the defining property, then write out every object that satisfies it.
    (i)
    The odd natural numbers form an unending list:
    \[\mathrm{A}=\{1,3,5,7,9,\ldots\}\]
    (ii)
    We need integers strictly between \(\displaystyle -\tfrac{1}{2}\) and \(\displaystyle \tfrac{9}{2}=4.5\). The integers in that range are \(\displaystyle 0,1,2,3,4\):
    \[\mathrm{B}=\{0,1,2,3,4\}\]
    (iii)
    \(\displaystyle x^{2} \leq 4\) means \(\displaystyle -2 \leq x \leq 2\), and \(\displaystyle x\) is an integer:
    \[\mathrm{C}=\{-2,-1,0,1,2\}\]
    (iv)
    LOYAL uses the letters L, O, Y, A, L; each distinct letter is written once:
    \[\mathrm{D}=\{\mathrm{L},\mathrm{O},\mathrm{Y},\mathrm{A}\}\]
    (v)
    The months with \(\displaystyle 31\) days are January, March, May, July, August, October, December; the rest are:
    \[\mathrm{E}=\{\text{February},\text{April},\text{June},\text{September},\text{November}\}\]
    (vi)
    The letters before \(\displaystyle k\) are \(\displaystyle a,b,c,d,e,f,g,h,i,j\); removing the vowels \(\displaystyle a, e, i\) leaves the consonants:
    \[\mathrm{F}=\{b,c,d,f,g,h,j\}\]
    A = {$\displaystyle 1$, $\displaystyle 3$, $\displaystyle 5$, $\displaystyle 7$, $\displaystyle 9$, …}; B = {$\displaystyle 0$, $\displaystyle 1$, $\displaystyle 2$, $\displaystyle 3$, $\displaystyle 4$}; C = {−$\displaystyle 2$, −$\displaystyle 1$, $\displaystyle 0$, $\displaystyle 1$, $\displaystyle 2$}; D = {L, O, Y, A}; E = {February, April, June, September, November}; F = {b, c, d, f, g, h, j}

  6. Exercise 6

    Match each of the set on the left in the roster form with the same set on the right described in set-builder form:
    (a)
    {x:x\displaystyle \{x: x is a prime number and a divisor of 6\displaystyle 6}\displaystyle \}
    (b)
    {x:x\displaystyle \{x: x is an odd natural number less than 10\displaystyle 10}\displaystyle \}
    (c)
    {x:x\displaystyle \{x: x is natural number and divisor of 6\displaystyle 6}\displaystyle \}
    (d)
    {x:x\displaystyle \{x: x is a letter of the word MATHEMATICS }\displaystyle \}.
    (i)
    {1,2,3,6}\displaystyle \{1,2,3,6\}
    (ii)
    {2,3}\displaystyle \{2,3\}
    (iii)
    \{M,A,T,H,E,I,C,S\}
    (iv)
    {1,3,5,7,9}\displaystyle \{1,3,5,7,9\}

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    NCERT’s answer
    (i)
    ↔ (c) (ii) ↔ (a) (iii) ↔ (d) (iv) ↔ (b)
    Convert each set-builder description to roster form, then match.
    (a)
    The divisors of \(\displaystyle 6\) are \(\displaystyle 1,2,3,6\); of these the primes are \(\displaystyle 2\) and \(\displaystyle 3\). So the set is \(\displaystyle \{2,3\}\) — this is (ii).
    (b)
    The odd natural numbers less than \(\displaystyle 10\) are \(\displaystyle 1,3,5,7,9\). So the set is \(\displaystyle \{1,3,5,7,9\}\) — this is (iv).
    (c)
    The natural numbers dividing \(\displaystyle 6\) are \(\displaystyle 1,2,3,6\). So the set is \(\displaystyle \{1,2,3,6\}\) — this is (i).
    (d)
    MATHEMATICS uses the letters M, A, T, H, E, M, A, T, I, C, S; the distinct letters are M, A, T, H, E, I, C, S — this is (iii).
    (a) ↔ (ii), (b) ↔ (iv), (c) ↔ (i), (d) ↔ (iii)