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EXERCISE 1.2 1–6 (part 2 of 6)

  1. Exercise 1

    Which of the following are examples of the null set
    (i)
    Set of odd natural numbers divisible by 2\displaystyle 2
    (ii)
    Set of even prime numbers
    (iii)
    {x:x\displaystyle \{x: x is a natural numbers, x<5\displaystyle x<5 and x>7}\displaystyle x>7\}
    (iv)
    {y:y\displaystyle \{y: y is a point common to any two parallel lines }\displaystyle \}

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    NCERT’s answer
    (i)
    , (iii), (iv)
    Null-set test. A set is the null (empty) set \(\displaystyle \phi\) when no object satisfies its defining property. So look for a description that is impossible to fulfil.
    (i)
    An odd number leaves remainder \(\displaystyle 1\) on division by \(\displaystyle 2\), so it can never be divisible by \(\displaystyle 2\). No such number exists — null set.
    (ii)
    \(\displaystyle 2\) is even and prime, so the set is \(\displaystyle \{2\}\); it has one element — not a null set.
    (iii)
    No number can satisfy \(\displaystyle x<5\) and \(\displaystyle x>7\) at the same time — null set.
    (iv)
    Parallel lines, by definition, never meet, so they have no point in common — null set.
    Null sets: (i), (iii) and (iv). (ii) is not a null set, since it equals {$\displaystyle 2$}.

  2. Exercise 2

    Which of the following sets are finite or infinite
    (i)
    The set of months of a year
    (ii)
    {1,2,3,…}\displaystyle \{1,2,3, \ldots\}
    (iii)
    {1,2,3,…99,100}\displaystyle \{1,2,3, \ldots 99,100\}
    (iv)
    The set of positive integers greater than 100\displaystyle 100
    (v)
    The set of prime numbers less than 99\displaystyle 99

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    Finite or infinite. A set is finite if its elements can be counted and the counting comes to an end; otherwise it is infinite.
    (i)
    A year has exactly \(\displaystyle 12\) months — finite.
    (ii)
    \(\displaystyle \{1,2,3,\ldots\}\) is the set of all natural numbers, and the list never ends — infinite.
    (iii)
    \(\displaystyle \{1,2,3,\ldots,99,100\}\) has exactly \(\displaystyle 100\) elements — finite.
    (iv)
    The integers \(\displaystyle 101,102,103,\ldots\) go on for ever — infinite.
    (v)
    The primes below \(\displaystyle 99\) are a part of the \(\displaystyle 98\) numbers \(\displaystyle 1,2,\ldots,98\), so the counting stops — finite.
    Finite: (i), (iii), (v). Infinite: (ii), (iv).

  3. Exercise 3

    State whether each of the following set is finite or infinite:
    (i)
    The set of lines which are parallel to the x\displaystyle x-axis
    (ii)
    The set of letters in the English alphabet
    (iii)
    The set of numbers which are multiple of 5\displaystyle 5
    (iv)
    The set of animals living on the earth
    (v)
    The set of circles passing through the origin (0,0)\displaystyle (0,0)

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    Finite or infinite. Ask whether the counting of elements comes to an end.
    (i)
    One line parallel to the \(\displaystyle x\)-axis passes through each point of the \(\displaystyle y\)-axis, and there are endlessly many such points — infinite.
    (ii)
    The English alphabet has exactly \(\displaystyle 26\) letters — finite.
    (iii)
    The multiples \(\displaystyle 5,10,15,\ldots\) never stop — infinite.
    (iv)
    The animals alive on the earth are enormously many, but they can in principle be counted — finite.
    (v)
    A circle through the origin is fixed by choosing its centre, and the centre may be any point of the plane, so there are endlessly many — infinite.
    Finite: (ii), (iv). Infinite: (i), (iii), (v).

  4. Exercise 4

    In the following, state whether A=B\displaystyle \mathrm{A}=\mathrm{B} or not:
    (i)
    A={a,b,c,d},B={d,c,b,a}\displaystyle \mathrm{A}=\{a, b, c, d\}, \mathrm{B}=\{d, c, b, a\}
    (ii)
    A={4,8,12,16}B={8,4,16,18}\displaystyle \mathrm{A}=\{4,8,12,16\} \mathrm{B}=\{8,4,16,18\}
    (iii)
    A={2,4,6,8,10}B={x:x\displaystyle \mathrm{A}=\{2,4,6,8,10\} \quad \mathrm{B}=\{x: x is positive even integer and x≤10}\displaystyle x \leq 10\}
    (iv)
    A={x:x\displaystyle \mathrm{A}=\{x: x is a multiple of 10\displaystyle 10},B={10,15,20,25,30,…}\displaystyle \}, \quad \mathrm{B}=\{10,15,20,25,30, \ldots\}

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    NCERT’s answer
    (i)
    Yes (ii) No (iii) Yes (iv) No
    Equality of sets. \(\displaystyle \mathrm{A}=\mathrm{B}\) when every element of \(\displaystyle \mathrm{A}\) lies in \(\displaystyle \mathrm{B}\) and every element of \(\displaystyle \mathrm{B}\) lies in \(\displaystyle \mathrm{A}\). The order in which elements are written, and any repetition, make no difference.
    (i)
    \(\displaystyle \mathrm{A}=\{a,b,c,d\}\), \(\displaystyle \mathrm{B}=\{d,c,b,a\}\): the same four letters, only reordered. A = B.
    (ii)
    \(\displaystyle \mathrm{A}=\{4,8,12,16\}\), \(\displaystyle \mathrm{B}=\{8,4,16,18\}\): \(\displaystyle 12 \in \mathrm{A}\) but \(\displaystyle 12 \notin \mathrm{B}\). A ≠ B.
    (iii)
    The positive even integers with \(\displaystyle x \leq 10\) are \(\displaystyle 2,4,6,8,10\), so \(\displaystyle \mathrm{B}=\{2,4,6,8,10\}=\mathrm{A}\). A = B.
    (iv)
    \(\displaystyle \mathrm{A}=\{10,20,30,40,\ldots\}\), while \(\displaystyle 15 \in \mathrm{B}\) and \(\displaystyle 15\) is not a multiple of \(\displaystyle 10\). A ≠ B.
    A = B in (i) and (iii); A ≠ B in (ii) and (iv).

  5. Exercise 5

    Are the following pair of sets equal ? Give reasons.
    (i)
    A={2,3},B={x:x\displaystyle \mathrm{A}=\{2,3\}, \quad \mathrm{B}=\left\{x: x\right. is solution of x2+5x+6=0}\displaystyle \left.x^{2}+5 x+6=0\right\}
    (ii)
    A={x:x\displaystyle \mathrm{A}=\{x: x is a letter in the word FOLLOW }\displaystyle \} B={y:y\displaystyle \mathrm{B}=\{y: y is a letter in the word WOLF }\displaystyle \}

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    NCERT’s answer
    (i)
    No (ii) Yes
    Equality of sets. Two sets are equal only if they contain exactly the same elements, so first put each set into roster form.
    (i)
    Solve \(\displaystyle x^{2}+5x+6=0\):
    \[x^{2}+5x+6=(x+2)(x+3)=0 \quad\Longrightarrow\quad x=-2 \text{ or } x=-3\]
    So \(\displaystyle \mathrm{B}=\{-2,-3\}\), while \(\displaystyle \mathrm{A}=\{2,3\}\). Since \(\displaystyle 2 \notin \mathrm{B}\), the sets are not equal.
    (ii)
    FOLLOW uses the letters F, O, L, L, O, W, so \(\displaystyle \mathrm{A}=\{\mathrm{F},\mathrm{O},\mathrm{L},\mathrm{W}\}\). WOLF gives \(\displaystyle \mathrm{B}=\{\mathrm{W},\mathrm{O},\mathrm{L},\mathrm{F}\}\). These are the same four letters, so the sets are equal.
    (i) A ≠ B, because B = {−$\displaystyle 2$, −$\displaystyle 3$}. (ii) A = B = {F, O, L, W}.

  6. Exercise 6

    From the sets given below, select equal sets : A={2,4,8,12},B={1,2,3,4},C={4,8,12,14},D={3,1,4,2}E={−1,1},F={0,a},G={1,−1},H={0,1}\begin{array}{llll} \mathrm{A}=\{2,4,8,12\}, & \mathrm{B}=\{1,2,3,4\}, & \mathrm{C}=\{4,8,12,14\}, & \mathrm{D}=\{3,1,4,2\} \\ \mathrm{E}=\{-1,1\}, & \mathrm{F}=\{0, a\}, & \mathrm{G}=\{1,-1\}, & \mathrm{H}=\{0,1\} \end{array}

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    Equality of sets. Compare the sets element by element; order and repetition are irrelevant.\(\displaystyle \mathrm{B}=\{1,2,3,4\}\) and \(\displaystyle \mathrm{D}=\{3,1,4,2\}\) contain exactly the numbers \(\displaystyle 1,2,3,4\), so \(\displaystyle \mathrm{B}=\mathrm{D}\).\(\displaystyle \mathrm{E}=\{-1,1\}\) and \(\displaystyle \mathrm{G}=\{1,-1\}\) contain exactly \(\displaystyle -1\) and \(\displaystyle 1\), so \(\displaystyle \mathrm{E}=\mathrm{G}\).No other pair matches:
    \(\displaystyle \mathrm{A}=\{2,4,8,12\}\) and \(\displaystyle \mathrm{C}=\{4,8,12,14\}\) differ, since \(\displaystyle 2 \in \mathrm{A}\) but \(\displaystyle 2 \notin \mathrm{C}\).
    \(\displaystyle \mathrm{F}=\{0,a\}\) and \(\displaystyle \mathrm{H}=\{0,1\}\) differ, since \(\displaystyle a\) is a letter and not the number \(\displaystyle 1\).
    B = D and E = G