Q1. Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) { x : x is a natural numbers, x < 5 and x> 7 }
(iv) { y : y is a point common to any two parallel lines}

Solution

Solution: A null set is a set which contains no elements. We examine each given collection to check whether it has any element or not.

\( \begin{aligned} \text{(i)}\;& \text{The set of odd natural numbers divisible by } 2. \\ &\text{No odd natural number is divisible by } 2. \\ &\Rightarrow \text{This set has no element and is a null set.} \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& \text{The set of even prime numbers.} \\ &\text{The number } 2 \text{ is an even prime number.} \\ &\Rightarrow \text{This set contains one element and is not a null set.} \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& \{x : x \text{ is a natural number},\; x < 5 \text{ and } x> 7\}. \\ &\text{There is no natural number which is simultaneously less than } 5 \text{ and greater than } 7. \\ &\Rightarrow \text{This set is a null set.} \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& \{y : y \text{ is a point common to any two parallel lines}\}. \\ &\text{Parallel lines never intersect and hence have no common point.} \\ &\Rightarrow \text{This set is a null set.} \end{aligned} \)

Hence, the examples of the null set are: (i), (iii) and (iv).

Q2. Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3, . . .}
(iii) {1, 2, 3, . . .99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99

Solution

Solution: A set is called finite if it has a definite and limited number of elements, whereas a set is infinite if its elements cannot be counted completely and continue without end.

\( \begin{aligned} \text{(i)}\;& \text{The set of months of a year.} \\ &\text{A year has exactly } 12 \text{ months.} \\ &\Rightarrow \text{This set has a fixed number of elements and is a finite set.} \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& \{1, 2, 3, \ldots\}. \\ &\text{This set represents all natural numbers starting from } 1. \\ &\text{There is no last natural number.} \\ &\Rightarrow \text{This set is infinite.} \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& \{1, 2, 3, \ldots, 99, 100\}. \\ &\text{The elements start from } 1 \text{ and end at } 100. \\ &\Rightarrow \text{This set contains a definite number of elements and is finite.} \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& \text{The set of positive integers greater than } 100. \\ &\text{The numbers } 101, 102, 103, \ldots \text{ continue without end.} \\ &\Rightarrow \text{This set is infinite.} \end{aligned} \)

\( \begin{aligned} \text{(v)}\;& \text{The set of prime numbers less than } 99. \\ &\text{There are only a limited number of prime numbers less than } 99. \\ &\Rightarrow \text{This set is finite.} \end{aligned} \)

Q3. State whether each of the following set is finite or infinite:

Solution

Solution: A set is said to be finite if it contains a definite and countable number of elements, and infinite if its elements continue without end and cannot be completely counted.

\( \begin{aligned} \text{(i)}\;& \text{The set of lines which are parallel to the x-axis.} \\ &\text{Each line parallel to the x-axis has an equation of the form } y = c, \text{ where } c \in \mathbb{R}. \\ &\text{Since } c \text{ can take infinitely many real values,} \\ &\Rightarrow \text{this set is infinite.} \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& \text{The set of letters in the English alphabet.} \\ &\text{There are exactly } 26 \text{ letters in the English alphabet.} \\ &\Rightarrow \text{this set is finite.} \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& \text{The set of numbers which are multiples of } 5. \\ &\text{The multiples } 5, 10, 15, 20, \ldots \text{ continue indefinitely.} \\ &\Rightarrow \text{this set is infinite.} \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& \text{The set of animals living on the earth.} \\ &\text{At any given time, the number of animals on earth is limited.} \\ &\Rightarrow \text{this set is finite.} \end{aligned} \)

\( \begin{aligned} \text{(v)}\;& \text{The set of circles passing through the origin } (0,0). \\ &\text{Infinitely many circles can be drawn passing through a fixed point.} \\ &\Rightarrow \text{this set is infinite.} \end{aligned} \)

Q4. In the following, state whether A = B or not:
\( \begin{array}{l} (i)\;A = \{ a, b, c, d \}& \;B = \{ d, c, b, a \}\\ (ii)\;A = \{ 4, 8, 12, 16 \} &\;B = \{ 8, 4, 16, 18\}\\ (iii)\;A = {2, 4, 6, 8, 10}&\;B = \{ x : x\text{ is positive even integer and }x ≤ 10\}\\ (iv)\;A = \{ x : x\text{ is a multiple of }10\},&\;B = \{ 10, 15, 20, 25, 30, . . . \} \end{array} \)

Solution

Solution: Two sets are said to be equal if and only if they contain exactly the same elements, irrespective of the order in which the elements are written.

\( \begin{aligned} \text{(i)}\;& A = \{a, b, c, d\}, \quad B = \{d, c, b, a\}. \\ &\text{Both sets contain the same elements, only the order is different.} \\ &\Rightarrow A = B. \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& A = \{4, 8, 12, 16\}, \quad B = \{8, 4, 16, 18\}. \\ &\text{The element } 12 \text{ belongs to } A \text{ but not to } B, \\ &\text{and } 18 \text{ belongs to } B \text{ but not to } A. \\ &\Rightarrow A \ne B. \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& A = \{2, 4, 6, 8, 10\}, \\ & B = \{x : x \text{ is a positive even integer and } x \le 10\}. \\ &\text{The set } B \text{ also represents the elements } \{2, 4, 6, 8, 10\}. \\ &\Rightarrow A = B. \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& A = \{x : x \text{ is a multiple of } 10\}, \\ & B = \{10, 15, 20, 25, 30, \ldots\}. \\ &\text{The set } A \text{ contains only multiples of } 10, \\ &\text{whereas } B \text{ contains numbers such as } 15 \text{ and } 25 \text{ which are not multiples of } 10. \\ &\Rightarrow A \ne B. \end{aligned} \)

Q5. Are the following pair of sets equal ? Give reasons.
\( \begin{array}{l} (i)\; A = \{2, 3\}&\;B = \{x : x\text{ is solution of }x^2 + 5x + 6 = 0\}\\ (ii)\; A = \{ x : x\text{ is a letter in the word FOLLOW}\}&\;B = \{ y : y\text{ is a letter in the word WOLF}\} \end{array} \)

Solution

Solution: Two sets are equal if they contain exactly the same elements, even if they are described differently or written in a different order.

\( \begin{aligned} \text{(i)}\;& A = \{2, 3\}, \quad B = \{x : x \text{ is a solution of } x^2 + 5x + 6 = 0\}. \\ &\text{Solving the quadratic equation,} \\ &x^2 + 5x + 6 = 0 \\ &\Rightarrow (x + 2)(x + 3) = 0 \\ &\Rightarrow x = -2 \text{ or } x = -3. \\ &\text{Thus, } B = \{-2, -3\}. \\ &\text{Since } A = \{2, 3\} \text{ and } B = \{-2, -3\} \text{ do not contain the same elements,} \\ &\Rightarrow A \ne B. \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& A = \{x : x \text{ is a letter in the word FOLLOW}\}, \\ & B = \{y : y \text{ is a letter in the word WOLF}\}. \\ &\text{The distinct letters in the word FOLLOW are } \{F, O, L, W\}. \\ &\text{The distinct letters in the word WOLF are also } \{W, O, L, F\}. \\ &\text{Both sets contain exactly the same letters.} \\ &\Rightarrow A = B. \end{aligned} \)

Q6.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}

Solution

Solution: Two sets are equal if they contain exactly the same elements, regardless of the order in which the elements are written.

\( \begin{aligned} A &= \{2, 4, 8, 12\}, \\ B &= \{1, 2, 3, 4\}, \\ C &= \{4, 8, 12, 14\}, \\ D &= \{3, 1, 4, 2\}. \end{aligned} \)

\( \begin{aligned} E &= \{-1, 1\}, \\ F &= \{0, a\}, \\ G &= \{1, -1\}, \\ H &= \{0, 1\}. \end{aligned} \)

On comparing the elements, the sets \(B = \{1, 2, 3, 4\}\) and \(D = \{3, 1, 4, 2\}\) contain exactly the same elements, only arranged in a different order. Hence, \(B = D\).

Similarly, the sets \(E = \{-1, 1\}\) and \(G = \{1, -1\}\) have the same elements written in a different order. Hence, \(E = G\).

All other sets differ from each other by at least one element and therefore are not equal.