Question 1(i)

The collection of all the months of a year beginning with the letter J.

Solution: The months are January, June, and July. This collection is well-defined because we can definitely decide whether a month begins with the letter J or not. Hence, it is a set.

Question 1(ii)

The collection of ten most talented writers of India.

Solution: This collection is not well-defined because the criteria for determining a writer's talent may vary from person to person. Hence, it is not a set.

Question 1(iii)

A team of eleven best-cricket batsmen of the world.

Solution: This collection is not well-defined because 'best' is subjective and the criteria for being the best batsman varies. Hence, it is not a set.

Question 1(iv)

The collection of all boys in your class.

Solution: This collection is well-defined because we can definitely identify all the boys present in a specific class. Hence, it is a set.

Question 1(v)

The collection of all natural numbers less than 100.

Solution: This is a well-defined collection as any number can be definitely tested to be a natural number less than 100. Hence, it is a set.

Question 1(vi)

A collection of novels written by the writer Munshi Prem Chand.

Solution: This is a well-defined collection because we can definitely list all novels written by Munshi Prem Chand. Hence, it is a set.

Question 1(vii)

The collection of all even integers.

Solution: This is a well-defined collection since we can clearly determine whether an integer is even or not. Hence, it is a set.

Question 1(viii)

The collection of questions in this Chapter.

Solution: This is a well-defined collection since the scope is restricted to the questions inside the textbook chapter. Hence, it is a set.

Question 1(ix)

A collection of most dangerous animals of the world.

Solution: This collection is not well-defined because the term 'dangerous' is subjective and varies from person to person. Hence, it is not a set.

Question 2

Let $A = \{1, 2, 3, 4, 5, 6\}$. Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces:

(i) $5 \dots A$
(ii) $8 \dots A$
(iii) $0 \dots A$
(iv) $4 \dots A$
(v) $2 \dots A$
(vi) $10 \dots A$

Solution:
(i) $5 \in A$ (since 5 is an element of A)
(ii) $8 \notin A$ (since 8 is not in A)
(iii) $0 \notin A$ (since 0 is not in A)
(iv) $4 \in A$ (since 4 is an element of A)
(v) $2 \in A$ (since 2 is an element of A)
(vi) $10 \notin A$ (since 10 is not in A)

Question 3

Write the following sets in roster form:

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

Solution:
(i) Integers from -3 up to less than 7 are $\{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6\}$.
(ii) Natural numbers less than 6 are $\{1, 2, 3, 4, 5\}$.
(iii) Two-digit numbers whose sum of digits is 8: $\{17, 26, 35, 44, 53, 62, 71, 80\}$.
(iv) Prime divisors of 60 ($60 = 2^2 \times 3 \times 5$): $\{2, 3, 5\}$.
(v) Letters of TRIGONOMETRY without repetition: $\{T, R, I, G, O, N, E, M, Y\}$.
(vi) Letters of BETTER without repetition: $\{B, E, T, R\}$.

Question 4

Write the following sets in the set-builder form:

(i) $\{3, 6, 9, 12\}$
(ii) $\{2, 4, 8, 16, 32\}$
(iii) $\{5, 25, 125, 625\}$
(iv) $\{2, 4, 6, \dots\}$
(v) $\{1, 4, 9, \dots, 100\}$

Solution:
(i) $\{x : x = 3n, n \in \mathbb{N} \text{ and } 1 \le n \le 4\}$
(ii) $\{x : x = 2^n, n \in \mathbb{N} \text{ and } 1 \le n \le 5\}$
(iii) $\{x : x = 5^n, n \in \mathbb{N} \text{ and } 1 \le n \le 4\}$
(iv) $\{x : x \text{ is an even natural number\}}$
(v) $\{x : x = n^2, n \in \mathbb{N} \text{ and } 1 \le n \le 10\}$

Question 5

List all the elements of the following sets:

(i) $A = \{x : x \text{ is an odd natural number\}}$
(ii) $B = \left\{x : x \text{ is an integer, } -\frac{1}{2} < x < \frac{9}{2}\right\}$
(iii) $C = \{x : x \text{ is an integer, } x^2 \le 4\}$
(iv) $D = \{x : x \text{ is a letter in the word "LOYAL"\}}$
(v) $E = \{x : x \text{ is a month of a year not having } 31 \text{ days\}}$
(vi) $F = \{x : x \text{ is a consonant in the English alphabet which precedes } k\}$

Solution:
(i) $A = \{1, 3, 5, 7, \dots\}$
(ii) Integers between $-0.5$ and $4.5$ are $\{0, 1, 2, 3, 4\}$.
(iii) Integers whose squares are $\le 4$ are $\{-2, -1, 0, 1, 2\}$.
(iv) Letters of LOYAL are $\{L, O, Y, A\}$.
(v) Months not having 31 days: February, April, June, September, November.
(vi) Consonants preceding k: $b, c, d, f, g, h, j$.

Question 6

Match each of the set on the left in roster form with the same set on the right described in set-builder form:

(i) $\{1, 2, 3, 6\}$ (a) $\{x : x \text{ is a prime number and a divisor of } 6\}$
(ii) $\{2, 3\}$ (b) $\{x : x \text{ is an odd natural number less than } 10\}$
(iii) $\{M, A, T, H, E, I, C, S\}$ (c) $\{x : x \text{ is natural number and divisor of } 6\}$
(iv) $\{1, 3, 5, 7, 9\}$ (d) $\{x : x \text{ is a letter of the word MATHEMATICS\}$

Solution:
(i) Divisors of 6 are 1, 2, 3, 6. Matches (c).
(ii) Prime divisors of 6 are 2, 3. Matches (a).
(iii) Letters of MATHEMATICS are M, A, T, H, E, I, C, S. Matches (d).
(iv) Odd natural numbers less than 10 are 1, 3, 5, 7, 9. Matches (b).

Question 1

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 \text{ is a natural numbers, } x < 5 \text{ and } x > 7\}
(iv) $\{y : y \text{ is a point common to any two parallel lines\}$

Solution:
(i) Null set (no odd number is divisible by 2).
(ii) Not a null set (2 is an even prime number).
(iii) Null set (no natural number can be both less than 5 and greater than 7).
(iv) Null set (parallel lines never intersect, hence have no common point).

Question 2

Which of the following sets are finite or infinite:

(i) The set of months of a year
(ii) $\{1, 2, 3, \dots\}
(iii) $\{1, 2, 3, \dots, 99, 100\}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99

Solution:
(i) Finite (12 months).
(ii) Infinite (natural numbers continue indefinitely).
(iii) Finite (contains 100 elements).
(iv) Infinite (101, 102, 103, ... continues without end).
(v) Finite (bounded upper limit below 99).

Question 3

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

(i) The set of lines which are parallel to the $x$-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin $(0,0)$

Solution:
(i) Infinite (infinitely many parallel lines can be drawn).
(ii) Finite (26 letters).
(iii) Infinite (5, 10, 15, ... goes on forever).
(iv) Finite (though large, it is a countable definite number at any given time).
(v) Infinite (infinitely many circles can pass through the origin).

Question 4

In the following, state whether $A = B$ or not:

(i) $A = \{a, b, c, d\}$, $B = \{d, c, b, a\}$
(ii) $A = \{4, 8, 12, 16\}$, $B = \{8, 4, 16, 18\}$
(iii) $A = \{2, 4, 6, 8, 10\}$, $B = \{x : x \text{ is positive even integer and } x \le 10\}$
(iv) $A = \{x : x \text{ is a multiple of } 10\}$, $B = \{10, 15, 20, 25, 30, \dots\}$

Solution:
(i) $A = B$ (order of elements does not matter).
(ii) $A \neq B$ (18 is in B but not in A).
(iii) $A = B$ ($B = \{2, 4, 6, 8, 10\}$, which matches $A$).
(iv) $A \neq B$ (15 is in B, but 15 is not a multiple of 10).

Question 5

Are the following pairs of sets equal? Give reasons.

(i) $A = \{2, 3\}$, $B = \{x : x \text{ is solution of } x^2 + 5x + 6 = 0\}$
(ii) $A = \{x : x \text{ is a letter in the word FOLLOW\}$, $B = \{y : y \text{ is a letter in the word WOLF\}$

Solution:
(i) $x^2 + 5x + 6 = 0 \implies (x+2)(x+3) = 0 \implies x = -2, -3$. Thus $B = \{-2, -3\} \neq A$. Hence not equal.
(ii) $A = \{F, O, L, W\}$ and $B = \{W, O, L, F\}$. They have the exact same elements, so $A = B$.

Question 6

From the sets given below, select equal sets:

$A = \{2, 4, 8, 12\}$, $B = \{1, 2, 3, 4\}$, $C = \{4, 8, 12, 14\}$, $D = \{3, 1, 4, 2\}$
$E = \{-1, 1\}$, $F = \{0, a\}$, $G = \{1, -1\}$, $H = \{0, 1\}$

Solution:
$B = D = \{1, 2, 3, 4\}$
$E = G = \{-1, 1\}$

Question 1

Make correct statements by filling in the symbols $\subset$ or $\not\subset$ in the blank spaces:

(i) $\{2, 3, 4\} \dots \{1, 2, 3, 4, 5\}$
(ii) $\{a, b, c\} \dots \{b, c, d\}$
(iii) $\{x : x \text{ is a student of Class XI of your school\}} \dots \{x : x \text{ student of your school\}}$
(iv) $\{x : x \text{ is a circle in the plane\}} \dots \{x : x \text{ is a circle in the same plane with radius 1 unit\}}$
(v) $\{x : x \text{ is a triangle in a plane\}} \dots \{x : x \text{ is a rectangle in the plane\}}$
(vi) $\{x : x \text{ is an equilateral triangle in a plane\}} \dots \{x : x \text{ is a triangle in the same plane\}}$
(vii) $\{x : x \text{ is an even natural number\}} \dots \{x : x \text{ is an integer\}}$

Solution:
(i) $\{2, 3, 4\} \subset \{1, 2, 3, 4, 5\}$
(ii) $\{a, b, c\} \not\subset \{b, c, d\}$
(iii) $\{x : x \text{ is a student of Class XI of your school\}} \subset \{x : x \text{ student of your school\}}$
(iv) $\{x : x \text{ is a circle in the plane\}} \not\subset \{x : x \text{ is a circle in the same plane with radius 1 unit\}}$
(v) $\{x : x \text{ is a triangle in a plane\}} \not\subset \{x : x \text{ is a rectangle in the plane\}}$
(vi) $\{x : x \text{ is an equilateral triangle in a plane\}} \subset \{x : x \text{ is a triangle in the same plane\}}$
(vii) $\{x : x \text{ is an even natural number\}} \subset \{x : x \text{ is an integer\}}$

Question 2

Examine whether the following statements are true or false:

(i) $\{a, b\} \not\subset \{b, c, a\}$
(ii) $\{a, e\} \subset \{x : x \text{ is a vowel in the English alphabet\}}$
(iii) $\{1, 2, 3\} \subset \{1, 3, 5\}$
(iv) $\{a\} \subset \{a, b, c\}$
(v) $\{a\} \in \{a, b, c\}$
(vi) $\{x : x \text{ is an even natural number less than } 6\} \subset \{x : x \text{ is a natural number which divides } 36\}$

Solution:
(i) False ($\{a, b\}$ is a subset of $\{b, c, a\}$).
(ii) True (both $a$ and $e$ are vowels).
(iii) False (2 is not in the second set).
(iv) True.
(v) False (element of a set cannot be a subset unless it's a set, but here $\{a\}$ is not listed as an element).
(vi) True (even naturals less than 6 are 2, 4, both of which divide 36).

Question 3

Let $A = \{1, 2, \{3, 4\}, 5\}$. Which of the following statements are incorrect and why?

(i) $\{3, 4\} \subset A$
(ii) $\{3, 4\} \in A$
(iii) $\{\{3, 4\}\} \subset A$
(iv) $1 \in A$
(v) $1 \subset A$
(vi) $\{1, 2, 5\} \subset A$
(vii) $\{1, 2, 5\} \in A$
(viii) $\{1, 2, 3\} \subset A$
(ix) $\phi \in A$
(x) $\phi \subset A$
(xi) $\{\phi\} \subset A$

Solution:
(i) Incorrect, because $\{3, 4\}$ is an element of $A$, not a subset.
(ii) Correct, $\{3, 4\}$ is listed as an element.
(iii) Correct, $\{\{3, 4\}\}$ is a subset containing the element $\{3, 4\}$.
(iv) Correct, 1 is an element of $A$.
(v) Incorrect, 1 is an element, not a subset.
(vi) Correct, all elements 1, 2, 5 belong to $A$.
(vii) Incorrect, $\{1, 2, 5\}$ is a subset, not an element of $A$.
(viii) Incorrect, 3 is not an element of $A$.
(ix) Incorrect, $\phi$ is not an element of $A$.
(x) Correct, $\phi$ is a subset of every set.
(xi) Incorrect, $\phi$ is an element of $\{\phi\}$, and since $\phi \notin A$, $\{\phi\}$ is not a subset of $A$.

Question 4

Write down all the subsets of the following sets:

(i) $\{a\}$
(ii) $\{a, b\}$
(iii) $\{1, 2, 3\}$
(iv) $\phi$

Solution:
(i) $\phi, \{a\}$
(ii) $\phi, \{a\}, \{b\}, \{a, b\}$
(iii) $\phi, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{2, 3\}, \{1, 3\}, \{1, 2, 3\}$
(iv) $\phi$

Question 5

Write the following as intervals:

(i) $\{x : x \in \mathbb{R}, -4 < x \le 6\}$
(ii) $\{x : x \in \mathbb{R}, -12 < x < -10\}$
(iii) $\{x : x \in \mathbb{R}, 0 \le x < 7\}$
(iv) $\{x : x \in \mathbb{R}, 3 \le x \le 4\}$

Solution:
(i) $(-4, 6]$
(ii) $(-12, -10)$
(iii) $[0, 7)$
(iv) $[3, 4]$

Question 6

Write the following intervals in set-builder form:

(i) $(-3, 0)$
(ii) $[6, 12]$
(iii) $(6, 12]$
(iv) $[-23, 5)$

Solution:
(i) $\{x : x \in \mathbb{R}, -3 < x < 0\}$
(ii) $\{x : x \in \mathbb{R}, 6 \le x \le 12\}$
(iii) $\{x : x \in \mathbb{R}, 6 < x \le 12\}$
(iv) $\{x : x \in \mathbb{R}, -23 \le x < 5\}$

Question 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.

Solution:
(i) The set of all triangles in a plane.
(ii) The set of all triangles in a plane.

Question 8

Given the sets $A = \{1, 3, 5\}$, $B = \{2, 4, 6\}$ and $C = \{0, 2, 4, 6, 8\}$, which of the following may be considered as universal set(s) for all the three sets A, B and C:

(i) $\{0, 1, 2, 3, 4, 5, 6\}$
(ii) $\phi$
(iii) $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$
(iv) $\{1, 2, 3, 4, 5, 6, 7, 8\}$

Solution:
(i) Does not contain 8 (from set C).
(ii) Cannot contain any set.
(iii) Contains all elements of A, B, and C. Hence valid.
(iv) Does not contain 0 (from set C).

Question 1

Find the union of each of the following pairs of sets:

(i) $X = \{1, 3, 5\}$, $Y = \{1, 2, 3\}$
(ii) $A = [a, e, i, o, u\}$, $B = \{a, b, c\}$
(iii) $A = \{x : x \text{ is a natural number and multiple of } 3\}$, $B = \{x : x \text{ is a natural number less than } 6\}$
(iv) $A = \{x : x \text{ is a natural number and } 1 < x \le 6\}$, $B = \{x : x \text{ is a natural number and } 6 < x < 10\}$
(v) $A = \{1, 2, 3\}$, $B = \phi$

Solution:
(i) $X \cup Y = \{1, 2, 3, 5\}$
(ii) $A \cup B = \{a, b, c, e, i, o, u\}$
(iii) $A \cup B = \{1, 2, 4, 5, 3, 6, 9, 12, \dots\}$
(iv) $A \cup B = \{2, 3, 4, 5, 6, 7, 8, 9\}$
(v) $A \cup B = \{1, 2, 3\}$

Question 2

Let $A = \{a, b\}$, $B = \{a, b, c\}$. Is $A \subset B$? What is $A \cup B$?

Solution:
Yes, $A \subset B$ since every element of $A$ is in $B$.
$A \cup B = \{a, b, c\} = B$.

Question 3

If $A$ and $B$ are two sets such that $A \subset B$, then what is $A \cup B$?

Solution:
$A \cup B = B$.

Question 4

If $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5, 6\}$, $C = \{5, 6, 7, 8\}$ and $D = \{7, 8, 9, 10\}$; find:

(i) $A \cup B$
(ii) $A \cup C$
(iii) $B \cup C$
(iv) $B \cup D$
(v) $A \cup B \cup C$
(vi) $A \cup B \cup D$
(vii) $B \cup C \cup D$

Solution:
(i) $\{1, 2, 3, 4, 5, 6\}$
(ii) $\{1, 2, 3, 4, 5, 6, 7, 8\}$
(iii) $\{3, 4, 5, 6, 7, 8\}$
(iv) $\{3, 4, 5, 6, 7, 8, 9, 10\}$
(v) $\{1, 2, 3, 4, 5, 6, 7, 8\}$
(vi) $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$
(vii) $\{3, 4, 5, 6, 7, 8, 9, 10\}$

Question 5

Find the intersection of each pair of sets of question 1 above.

Solution:
(i) $X \cap Y = \{1, 3\}$
(ii) $A \cap B = \{a\}$
(iii) $A \cap B = \{3\}$
(iv) $A \cap B = \phi$
(v) $A \cap B = \phi$

Question 6

If $A = \{3, 5, 7, 9, 11\}$, $B = \{7, 9, 11, 13\}$, $C = \{11, 13, 15\}$ and $D = \{15, 17\}$; find:

(i) $A \cap B$
(ii) $B \cap C$
(iii) $A \cap C \cap D$
(iv) $A \cap C$
(v) $B \cap D$
(vi) $A \cap (B \cup C)$
(vii) $A \cap D$
(viii) $A \cap (B \cup D)$
(ix) $(A \cap B) \cap (B \cup C)$
(x) $(A \cup D) \cap (B \cup C)$

Solution:
(i) $\{7, 9, 11\}$
(ii) $\{11, 13\}$
(iii) $\phi$
(iv) $\{11\}$
(v) $\phi$
(vi) $B \cup C = \{7, 9, 11, 13, 15\}$, so $A \cap (B \cup C) = \{7, 9, 11\}$
(vii) $\phi$
(viii) $B \cup D = \{7, 9, 11, 13, 15, 17\}$, so $A \cap (B \cup D) = \{7, 9, 11\}$
(ix) $\{7, 9, 11\}$
(x) $\{7, 9, 11, 13, 15\}$

Question 7

If $A = \{x : x \text{ is a natural number\}}$, $B = \{x : x \text{ is an even natural number\}}$, $C = \{x : x \text{ is an odd natural number\}}$ and $D = \{x : x \text{ is a prime number\}}$, find:

(i) $A \cap B$
(ii) $A \cap C$
(iii) $A \cap D$
(iv) $B \cap C$
(v) $B \cap D$
(vi) $C \cap D$

Solution:
(i) $B$
(ii) $C$
(iii) $D$
(iv) $\phi$
(v) $\{2\}$
(vi) Set of odd primes $\{3, 5, 7, 11, \dots\}$

Question 8

Which of the following pairs of sets are disjoint:

(i) $\{1, 2, 3, 4\}$ and $\{x : x \text{ is a natural number and } 4 \le x \le 6\}$
(ii) $\{a, e, i, o, u\}$ and $\{c, d, e, f\}$
(iii) $\{x : x \text{ is an even integer\}}$ and $\{x : x \text{ is an odd integer\}}$

Solution:
(i) Not disjoint (4 is common).
(ii) Not disjoint (e is common).
(iii) Disjoint (no integer is both even and odd).

Question 9

If $A = \{3, 6, 9, 12, 15, 18, 21\}$, $B = \{4, 8, 12, 16, 20\}$, $C = \{2, 4, 6, 8, 10, 12, 14, 16\}$, $D = \{5, 10, 15, 20\}$; find:

(i) $A - B$
(ii) $A - C$
(iii) $A - D$
(iv) $B - A$
(v) $C - A$
(vi) $D - A$
(vii) $B - C$
(viii) $B - D$
(ix) $C - B$
(x) $D - B$
(xi) $C - D$
(xii) $D - C$

Solution:
(i) $\{3, 6, 9, 15, 18, 21\}$
(ii) $\{3, 9, 15, 21\}$
(iii) $\{3, 6, 9, 12, 18, 21\}$
(iv) $\{4, 8, 16, 20\}$
(v) $\{2, 8, 10, 14, 16\}$
(vi) $\{5, 10, 20\}$
(vii) $\{20\}$
(viii) $\{4, 8, 12, 16\}$
(ix) $\{2, 6, 10, 12, 14\}$
(x) $\{5, 15\}$
(xi) $\{2, 4, 6, 8, 12, 14, 16\}$
(xii) $\{5, 10, 20\}$

Question 10

If $X = \{a, b, c, d\}$ and $Y = \{f, b, d, g\}$, find:

(i) $X - Y$
(ii) $Y - X$
(iii) $X \cap Y$

Solution:
(i) $\{a, c\}$
(ii) $\{f, g\}$
(iii) $\{b, d\}$

Question 11

If $R$ is the set of real numbers and $Q$ is the set of rational numbers, then what is $R - Q$?

Solution:
The set of irrational numbers $T$.

Question 12

State whether each of the following statement is true or false. Justify your answer.

(i) $\{2, 3, 4, 5\}$ and $\{3, 6\}$ are disjoint sets.
(ii) $\{a, e, i, o, u\}$ and $\{a, b, c, d\}$ are disjoint sets.
(iii) $\{2, 6, 10, 14\}$ and $\{3, 7, 11, 15\}$ are disjoint sets.
(iv) $\{2, 6, 10\}$ and $\{3, 7, 11\}$ are disjoint sets.

Solution:
(i) False (3 is common).
(ii) False (a is common).
(iii) True (no common elements).
(iv) True (no common elements).

Question 1

Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $A = \{1, 2, 3, 4\}$, $B = \{2, 4, 6, 8\}$ and $C = \{3, 4, 5, 6\}$. Find:

(i) $A'$
(ii) $B'$
(iii) $(A \cup C)'$
(iv) $(A \cup B)'$
(v) $(A')'$
(vi) $(B - C)'$

Solution:
(i) $A' = \{5, 6, 7, 8, 9\}$
(ii) $B' = \{1, 3, 5, 7, 9\}$
(iii) $A \cup C = \{1, 2, 3, 4, 5, 6\}$, so $(A \cup C)' = \{7, 8, 9\}$
(iv) $A \cup B = \{1, 2, 3, 4, 6, 8\}$, so $(A \cup B)' = \{5, 7, 9\}$
(v) $(A')' = A = \{1, 2, 3, 4\}$
(vi) $B - C = \{2, 8\}$, so $(B - C)' = \{1, 3, 4, 5, 6, 7, 9\}$

Question 2

If $U = \{a, b, c, d, e, f, g, h\}$, find the complements of the following sets:

(i) $A = \{a, b, c\}$
(ii) $B = \{d, e, f, g\}$
(iii) $C = \{a, c, e, g\}$
(iv) $D = \{f, g, h, a\}$

Solution:
(i) $A' = \{d, e, f, g, h\}$
(ii) $B' = \{a, b, c, h\}$
(iii) $C' = \{b, d, f, h\}$
(iv) $D' = \{b, c, d, e\}$

Question 3

Taking the set of natural numbers as the universal set, write down the complements of the following sets:

(i) $\{x : x \text{ is an even natural number\}}$
(ii) $\{x : x \text{ is an odd natural number\}}$
(iii) $\{x : x \text{ is a positive multiple of } 3\}$
(iv) $\{x : x \text{ is a prime number\}}$
(v) $\{x : x \text{ is a natural number divisible by } 3 \text{ and } 5\}$
(vi) $\{x : x \text{ is a perfect square\}}$
(vii) $\{x : x \text{ is a perfect cube\}}$
(viii) $\{x : x + 5 = 8\}$
(ix) $\{x : 2x + 5 = 9\}$
(x) $\{x : x \ge 7\}$
(xi) $\{x : x \in \mathbb{N} \text{ and } 2x + 1 > 10\}$

Solution:
(i) Set of odd natural numbers.
(ii) Set of even natural numbers.
(iii) $\{x : x \in \mathbb{N}, x \text{ is not a multiple of } 3\}$
(iv) $\{1\} \cup \{x : x \in \mathbb{N}, x \text{ is a composite number\}}$
(v) $\{x : x \in \mathbb{N}, x \text{ is not divisible by both } 3 \text{ and } 5\}$
(vi) $\{x : x \in \mathbb{N}, x \text{ is not a perfect square\}}$
(vii) $\{x : x \in \mathbb{N}, x \text{ is not a perfect cube\}}$
(viii) $\{x \in \mathbb{N} : x \neq 3\}$
(ix) $\{x \in \mathbb{N} : x \neq 2\}$
(x) $\{1, 2, 3, 4, 5, 6\}$
(xi) $\{1, 2, 3, 4\}$

Question 4

If $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $A = \{2, 4, 6, 8\}$ and $B = \{2, 3, 5, 7\}$. Verify that:

(i) $(A \cup B)' = A' \cap B'$
(ii) $(A \cap B)' = A' \cup B'$

Solution:
(i) $A \cup B = \{2, 3, 4, 5, 6, 7, 8\}$, so $(A \cup B)' = \{1, 9\}$.
$A' = \{1, 3, 5, 7, 9\}$, $B' = \{1, 4, 6, 8, 9\}$.
$A' \cap B' = \{1, 9\}$. Hence verified.
(ii) $A \cap B = \{2\}$, so $(A \cap B)' = \{1, 3, 4, 5, 6, 7, 8, 9\}$.
$A' \cup B' = \{1, 3, 4, 5, 6, 7, 8, 9\}$. Hence verified.

Question 5

Draw appropriate Venn diagram for each of the following:

(i) $(A \cup B)'$
(ii) $A' \cap B'$
(iii) $(A \cap B)'$
(iv) $A' \cup B'$

Solution:
(i) Represents the region outside both circles $A$ and $B$.
(ii) Same as $(A \cup B)'$, region outside circles $A$ and $B$.
(iii) Represents the region outside the intersection of $A$ and $B$.
(iv) Same as $(A \cap B)'$, region outside the intersection of $A$ and $B$.

Question 6

Let $U$ be the set of all triangles in a plane. If $A$ is the set of all triangles with at least one angle different from $60^\circ$, what is $A'$?

Solution:
$A'$ is the set of all triangles where no angle is different from $60^\circ$, which means the set of all equilateral triangles.

Question 7

Fill in the blanks to make each of the following a true statement:

(i) $A \cup A' = \dots$
(ii) $\phi' \cap A = \dots$
(iii) $A \cap A' = \dots$
(iv) $U' \cap A = \dots$

Solution:
(i) $U$
(ii) $A$
(iii) $\phi$
(iv) $\phi$