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NCERT Solutions · Class 11 Mathematics Sets

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EXERCISE 1.3 1–8 (part 3 of 6)

  1. Exercise 1

    Make correct statements by filling in the symbols ⊂\displaystyle \subset or ⊄\displaystyle \not \subset in the blank spaces :
    (i)
    {2,3,4}…{1,2,3,4,5}\displaystyle \{2,3,4\} \ldots\{1,2,3,4,5\}
    (ii)
    {a,b,c}…{b,c,d}\displaystyle \{a, b, c\} \ldots\{b, c, d\}
    (iii)
    {x:x\displaystyle \{x: x is a student of Class XI of your school }…{x:x\displaystyle \} \ldots\{x: x student of your school }\displaystyle \}
    (iv)
    {x:x\displaystyle \{x: x is a circle in the plane }…{x:x\displaystyle \} \ldots\{x: x is a circle in the same plane with radius 1\displaystyle 1 unit\}
    (v)
    {x:x\displaystyle \{x: x is a triangle in a plane }…{x:x\displaystyle \} \ldots\{x: x is a rectangle in the plane }\displaystyle \}
    (vi)
    {x:x\displaystyle \{x: x is an equilateral triangle in a plane }…{x:x\displaystyle \} \ldots\{x: x is a triangle in the same plane }\displaystyle \}
    (vii)
    {x:x\displaystyle \{x: x is an even natural number }…{x:x\displaystyle \} \ldots\{x: x is an integer }\displaystyle \}

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    NCERT’s answer
    (i)
    $\displaystyle \subset$ (ii) $\displaystyle \not \subset$ (iii) ᄃ (iv) $\displaystyle \not \subset$ (v) $\displaystyle \not \subset$ (vi) ᄃ (vii) $\displaystyle \subset$
    Subset test. \(\displaystyle \mathrm{P} \subset \mathrm{Q}\) means every element of \(\displaystyle \mathrm{P}\) is also an element of \(\displaystyle \mathrm{Q}\). To show \(\displaystyle \mathrm{P} \not\subset \mathrm{Q}\) it is enough to exhibit one element of \(\displaystyle \mathrm{P}\) that is missing from \(\displaystyle \mathrm{Q}\).
    (i)
    \(\displaystyle 2,3,4\) all appear in \(\displaystyle \{1,2,3,4,5\}\): \(\displaystyle \{2,3,4\} \subset \{1,2,3,4,5\}\).
    (ii)
    \(\displaystyle a \notin \{b,c,d\}\): \(\displaystyle \{a,b,c\} \not\subset \{b,c,d\}\).
    (iii)
    Every Class XI student of your school is a student of your school: \(\displaystyle \subset\).
    (iv)
    A circle of radius \(\displaystyle 2\) units lies in the plane but is not of radius \(\displaystyle 1\) unit: \(\displaystyle \not\subset\).
    (v)
    A triangle is never a rectangle: \(\displaystyle \not\subset\).
    (vi)
    An equilateral triangle is a triangle: \(\displaystyle \subset\).
    (vii)
    Every even natural number is an integer: \(\displaystyle \subset\).
    (i) ⊂ (ii) ⊄ (iii) ⊂ (iv) ⊄ (v) ⊄ (vi) ⊂ (vii) ⊂

  2. Exercise 2

    Examine whether the following statements are true or false:
    (i)
    {a,b}⊄{b,c,a}\displaystyle \{a, b\} \not \subset\{b, c, a\}
    (ii)
    {a,e}⊂{x:x\displaystyle \{a, e\} \subset\{x: x is a vowel in the English alphabet }\displaystyle \}
    (iii)
    {1,2,3}⊂{1,3,5}\displaystyle \{1,2,3\} \subset\{1,3,5\}
    (iv)
    {a}⊂{a,b,c}\displaystyle \{a\} \subset\{a, b, c\}
    (v)
    {a}∈{a,b,c}\displaystyle \{a\} \in\{a, b, c\}
    (vi)
    {x:x\displaystyle \{x: x is an even natural number less than 6\displaystyle 6}⊂{x:x\displaystyle \} \subset\{x: x is a natural number which divides 36\displaystyle 36\}

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    NCERT’s answer
    (i)
    False (ii) True (iii) False (iv) True (v) False (vi) True
    Distinguish ⊂ from ∈. \(\displaystyle \mathrm{P} \subset \mathrm{Q}\) compares two sets and asks whether every element of \(\displaystyle \mathrm{P}\) lies in \(\displaystyle \mathrm{Q}\); \(\displaystyle x \in \mathrm{Q}\) asks whether the single object \(\displaystyle x\) is one of the listed elements of \(\displaystyle \mathrm{Q}\).
    (i)
    Both \(\displaystyle a\) and \(\displaystyle b\) belong to \(\displaystyle \{b,c,a\}\), so \(\displaystyle \{a,b\}\) is a subset. The statement \(\displaystyle \{a,b\} \not\subset \{b,c,a\}\) is therefore false.
    (ii)
    The vowels are \(\displaystyle a,e,i,o,u\), and both \(\displaystyle a\) and \(\displaystyle e\) are among them — true.
    (iii)
    \(\displaystyle 2 \in \{1,2,3\}\) but \(\displaystyle 2 \notin \{1,3,5\}\) — false.
    (iv)
    The only element of \(\displaystyle \{a\}\) is \(\displaystyle a\), and \(\displaystyle a \in \{a,b,c\}\) — true.
    (v)
    The elements of \(\displaystyle \{a,b,c\}\) are the letters \(\displaystyle a, b, c\), not the set \(\displaystyle \{a\}\) — false.
    (vi)
    The left set is \(\displaystyle \{2,4\}\); the divisors of \(\displaystyle 36\) are \(\displaystyle \{1,2,3,4,6,9,12,18,36\}\), which contains both \(\displaystyle 2\) and \(\displaystyle 4\) — true.
    True: (ii), (iv), (vi). False: (i), (iii), (v).

  3. Exercise 3

    Let A={1,2,{3,4},5}\displaystyle \mathrm{A}=\{1,2,\{3,4\}, 5\}. Which of the following statements are incorrect and why?
    (i)
    {3,4}⊂A\displaystyle \{3,4\} \subset \mathrm{A}
    (ii)
    {3,4}∈A\displaystyle \{3,4\} \in \mathrm{A}
    (iii)
    {{3,4}}⊂A\displaystyle \{\{3,4\}\} \subset \mathrm{A}
    (iv)
    1∈ A\displaystyle 1 \in \mathrm{~A}
    (v)
    1⊂\displaystyle 1 \subset A
    (vi)
    {1,2,5}⊂A\displaystyle \{1,2,5\} \subset \mathrm{A}
    (vii)
    {1,2,5}∈A\displaystyle \{1,2,5\} \in \mathrm{A}
    (viii)
    {1,2,3}⊂A\displaystyle \{1,2,3\} \subset \mathrm{A}
    (ix)
    ϕ∈A\displaystyle \phi \in \mathrm{A}
    (x)
    ϕ⊂A\displaystyle \phi \subset \mathrm{A}
    (xi)
    {ϕ}⊂A\displaystyle \{\phi\} \subset \mathrm{A}

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    NCERT’s answer
    (i)
    as $\displaystyle \{3,4\} \in \mathrm{A}$, (v) as $\displaystyle 1 \in \mathrm{~A}$, (vii) as $\displaystyle \{1,2,5\} \subset \mathrm{A}$, (viii) as $\displaystyle 3 \notin \mathrm{~A}$, (ix) as $\displaystyle \phi \subset \mathrm{A}$, (xi) as $\displaystyle \phi \subset \mathrm{A}$,
    Read the elements of A first. \(\displaystyle \mathrm{A}=\{1,2,\{3,4\},5\}\) has exactly four elements: the numbers \(\displaystyle 1\), \(\displaystyle 2\), \(\displaystyle 5\), and the set \(\displaystyle \{3,4\}\). Note that \(\displaystyle 3\) and \(\displaystyle 4\) themselves are not elements of \(\displaystyle \mathrm{A}\). Throughout, \(\displaystyle \in\) links an object to a set, while \(\displaystyle \subset\) links a set to a set.
    (i)
    \(\displaystyle \{3,4\} \subset \mathrm{A}\) — incorrect. This would need \(\displaystyle 3 \in \mathrm{A}\) and \(\displaystyle 4 \in \mathrm{A}\), but neither is an element of \(\displaystyle \mathrm{A}\).
    (ii)
    \(\displaystyle \{3,4\} \in \mathrm{A}\) — correct. The set \(\displaystyle \{3,4\}\) is itself one of the four elements listed.
    (iii)
    \(\displaystyle \{\{3,4\}\} \subset \mathrm{A}\) — correct. Its only element is \(\displaystyle \{3,4\}\), and \(\displaystyle \{3,4\} \in \mathrm{A}\).
    (iv)
    \(\displaystyle 1 \in \mathrm{A}\) — correct; \(\displaystyle 1\) is listed.
    (v)
    \(\displaystyle 1 \subset \mathrm{A}\) — incorrect. \(\displaystyle 1\) is a number, not a set, so \(\displaystyle \subset\) cannot be used with it. (\(\displaystyle \{1\} \subset \mathrm{A}\) would be correct.)
    (vi)
    \(\displaystyle \{1,2,5\} \subset \mathrm{A}\) — correct; each of \(\displaystyle 1,2,5\) is an element of \(\displaystyle \mathrm{A}\).
    (vii)
    \(\displaystyle \{1,2,5\} \in \mathrm{A}\) — incorrect; the set \(\displaystyle \{1,2,5\}\) is not one of the four elements of \(\displaystyle \mathrm{A}\).
    (viii)
    \(\displaystyle \{1,2,3\} \subset \mathrm{A}\) — incorrect; \(\displaystyle 3 \notin \mathrm{A}\).
    (ix)
    \(\displaystyle \phi \in \mathrm{A}\) — incorrect; \(\displaystyle \phi\) is not one of the four elements listed.
    (x)
    \(\displaystyle \phi \subset \mathrm{A}\) — correct; the empty set is a subset of every set.
    (xi)
    \(\displaystyle \{\phi\} \subset \mathrm{A}\) — incorrect. This would need \(\displaystyle \phi \in \mathrm{A}\), which is false by (ix).
    Incorrect: (i), (v), (vii), (viii), (ix), (xi). Correct: (ii), (iii), (iv), (vi), (x).

  4. Exercise 4

    Write down all the subsets of the following sets
    (i)
    {a}\displaystyle \{a\} (ii) {a,b}\displaystyle \{a, b\}
    (iii)
    \{1\displaystyle 1, 2\displaystyle 2, 3\displaystyle 3\}
    (iv)
    ϕ\displaystyle \phi

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    NCERT’s answer
    (i)
    $\displaystyle \phi,\{a\}$ (ii) $\displaystyle \phi,\{a\},\{b\},\{a, b\}$ (iii) $\displaystyle \phi,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}$ (iv) $\displaystyle \phi$
    Listing subsets. A subset is formed by choosing, for each element, whether to include it or leave it out, so a set with \(\displaystyle n\) elements has \(\displaystyle 2^{n}\) subsets. Always remember \(\displaystyle \phi\) and the set itself.
    (i)
    \(\displaystyle \{a\}\) has \(\displaystyle 2^{1}=2\) subsets: \(\displaystyle \phi,\ \{a\}\).
    (ii)
    \(\displaystyle \{a,b\}\) has \(\displaystyle 2^{2}=4\) subsets: \(\displaystyle \phi,\ \{a\},\ \{b\},\ \{a,b\}\).
    (iii)
    \(\displaystyle \{1,2,3\}\) has \(\displaystyle 2^{3}=8\) subsets: \(\displaystyle \phi,\ \{1\},\ \{2\},\ \{3\},\ \{1,2\},\ \{1,3\},\ \{2,3\},\ \{1,2,3\}\).
    (iv)
    \(\displaystyle \phi\) has \(\displaystyle 2^{0}=1\) subset: \(\displaystyle \phi\) itself.
    (i) φ, {a} (ii) φ, {a}, {b}, {a, b} (iii) φ, {$\displaystyle 1$}, {$\displaystyle 2$}, {$\displaystyle 3$}, {$\displaystyle 1$, $\displaystyle 2$}, {$\displaystyle 1$, $\displaystyle 3$}, {$\displaystyle 2$, $\displaystyle 3$}, {$\displaystyle 1$, $\displaystyle 2$, $\displaystyle 3$} (iv) φ

  5. Exercise 5

    Write the following as intervals :
    (i)
    {x:x∈R,−4<x≤6}\displaystyle \{x: x \in \mathrm{R},-4<x \leq 6\}
    (ii)
    {x:x∈R,−12<x<−10}\displaystyle \{x: x \in \mathrm{R},-12<x<-10\}
    (iii)
    {x:x∈R,0≤x<7}\displaystyle \{x: x \in \mathrm{R}, 0 \leq x<7\}
    (iv)
    {x:x∈R,3≤x≤4}\displaystyle \{x: x \in \mathrm{R}, 3 \leq x \leq 4\}

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    NCERT’s answer
    (i)
    (- $\displaystyle 4$, $\displaystyle 6$] (ii) $\displaystyle (-12, -10)$ (iii) [$\displaystyle 0,7$) (iv) [ $\displaystyle 3$, $\displaystyle 4$ ]
    Interval notation. For real numbers, a round bracket excludes the end point (it goes with \(\displaystyle <\)) and a square bracket includes it (it goes with \(\displaystyle \leq\)).
    (i)
    \(\displaystyle -4<x \leq 6\): \(\displaystyle -4\) excluded, \(\displaystyle 6\) included — \(\displaystyle (-4,\,6]\).
    (ii)
    \(\displaystyle -12<x<-10\): both excluded — \(\displaystyle (-12,\,-10)\).
    (iii)
    \(\displaystyle 0 \leq x<7\): \(\displaystyle 0\) included, \(\displaystyle 7\) excluded — \(\displaystyle [0,\,7)\).
    (iv)
    \(\displaystyle 3 \leq x \leq 4\): both included — \(\displaystyle [3,\,4]\).
    (i) (−$\displaystyle 4$, $\displaystyle 6$] (ii) (−$\displaystyle 12$, −$\displaystyle 10$) (iii) [$\displaystyle 0$, $\displaystyle 7$) (iv) [$\displaystyle 3$, $\displaystyle 4$]

  6. Exercise 6

    Write the following intervals in set-builder form :
    (i)
    (−3,0)\displaystyle (-3, 0)
    (ii)
    [6\displaystyle 6, 12\displaystyle 12]
    (iii)
    (6,12]\displaystyle (6,12]
    (iv)
    [-23\displaystyle 23, 5\displaystyle 5)

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    Set-builder form of an interval. An interval is a set of real numbers, so write \(\displaystyle \{x : x \in \mathbf{R}, \ldots\}\). A round bracket becomes a strict inequality \(\displaystyle <\); a square bracket becomes \(\displaystyle \leq\).
    (i)
    \(\displaystyle (-3,0)\): both ends excluded —
    \[\{x : x \in \mathbf{R},\; -3<x<0\}\]
    (ii)
    \(\displaystyle [6,12]\): both ends included —
    \[\{x : x \in \mathbf{R},\; 6 \leq x \leq 12\}\]
    (iii)
    \(\displaystyle (6,12]\): \(\displaystyle 6\) excluded, \(\displaystyle 12\) included —
    \[\{x : x \in \mathbf{R},\; 6<x \leq 12\}\]
    (iv)
    \(\displaystyle [-23,5)\): \(\displaystyle -23\) included, \(\displaystyle 5\) excluded —
    \[\{x : x \in \mathbf{R},\; -23 \leq x<5\}\]
    (i) {x : x ∈ R, −$\displaystyle 3$ < x < $\displaystyle 0$} (ii) {x : x ∈ R, $\displaystyle 6$ ≤ x ≤ $\displaystyle 12$} (iii) {x : x ∈ R, $\displaystyle 6$ < x ≤ $\displaystyle 12$} (iv) {x : x ∈ R, −$\displaystyle 23$ ≤ x < $\displaystyle 5$}

  7. Exercise 7

    What universal set(s) would you propose for each of the following :
    (i)
    The set of right triangles.
    (ii)
    The set of isosceles triangles.

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    Universal set. A set \(\displaystyle \mathrm{U} \) can serve as a universal set for a given set \(\displaystyle \mathrm{A} \) only if \(\displaystyle \mathrm{A} \subset \mathrm{U} \), i.e. \(\displaystyle \mathrm{U} \) must contain every object being discussed. Here both sets consist of triangles, so the natural choice is a set that holds all triangles.
    (i)
    Every right triangle is a triangle, so take
    \[\mathrm{U} = \{x : x \text{ is a triangle in a plane}\}. \]
    Any larger collection also works — the set of all polygons, or the set of all plane figures.
    (ii)
    Every isosceles triangle is likewise a triangle, so the same choice serves:
    \[\mathrm{U} = \{x : x \text{ is a triangle in a plane}\}. \]
    The set of all triangles is a suitable universal set in both (i) and (ii).

  8. Exercise 8

    Given the sets A={1,3,5},B={2,4,6}\displaystyle \mathrm{A}=\{1,3,5\}, \mathrm{B}=\{2,4,6\} and C={0,2,4,6,8}\displaystyle \mathrm{C}=\{0,2,4,6,8\}, which of the following may be considered as universal set (s) for all the three sets A, B and C
    (i)
    {0,1,2,3,4,5,6}\displaystyle \{0,1,2,3,4,5,6\}
    (ii)
    ϕ\displaystyle \phi
    (iii)
    \{0,1,2,3,4,5,6,7,8,9,10\displaystyle 0,1,2,3,4,5,6,7,8,9,10\}
    (iv)
    {1,2,3,4,5,6,7,8}\displaystyle \{1,2,3,4,5,6,7,8\}

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    Test each candidate for containment. A set \(\displaystyle \mathrm{U} \) is a universal set for \(\displaystyle \mathrm{A}, \mathrm{B}, \mathrm{C} \) only if \(\displaystyle \mathrm{A} \subset \mathrm{U} \), \(\displaystyle \mathrm{B} \subset \mathrm{U} \) and \(\displaystyle \mathrm{C} \subset \mathrm{U} \) — equivalently, only if \(\displaystyle \mathrm{U} \) contains
    \[\mathrm{A} \cup \mathrm{B} \cup \mathrm{C} = \{0,1,2,3,4,5,6,8\}. \]
    (i)
    \(\displaystyle \{0,1,2,3,4,5,6\} \): does not contain \(\displaystyle 8 \in \mathrm{C} \), so \(\displaystyle \mathrm{C} \not\subset \mathrm{U} \). Not a universal set.
    (ii)
    \(\displaystyle \phi \): the empty set contains no element at all, so it cannot contain \(\displaystyle \mathrm{A} \), \(\displaystyle \mathrm{B} \) or \(\displaystyle \mathrm{C} \). Not a universal set.
    (iii)
    \(\displaystyle \{0,1,2,3,4,5,6,7,8,9,10\} \): it contains \(\displaystyle 0,1,2,3,4,5,6,8 \), hence \(\displaystyle \mathrm{A} \subset \mathrm{U} \), \(\displaystyle \mathrm{B} \subset \mathrm{U} \) and \(\displaystyle \mathrm{C} \subset \mathrm{U} \). This one works.
    (iv)
    \(\displaystyle \{1,2,3,4,5,6,7,8\} \): does not contain \(\displaystyle 0 \in \mathrm{C} \). Not a universal set.
    Only (iii) \(\displaystyle \{0,1,2,3,4,5,6,7,8,9,10\} \) can be taken as a universal set for A, B and C.