Sets

Q1: Which of the following collections of objects are sets?

i. All the months in a year.

Step 1: The collection of all months is well-defined and universally agreed.
Step 2: There are exactly 12 months, and they are fixed.
Answer: Yes, it is a set.

ii. All the rivers flowing in India.

Step 1: This collection is definite and can be listed.
Step 2: Hence, it is well-defined.
Answer: Yes, it is a set.

iii. All the planets in the solar system.

Step 1: The solar system has a fixed number of known planets.
Step 2: The collection is universally accepted.
Answer: Yes, it is a set.

iv. All the interesting dramas written by Shakespeare.

Step 1: The word “interesting” is subjective.
Step 2: What is interesting for one may not be for another.
Answer: No, it is not a set.

v. All the short boys of your class.

Step 1: The term “short” is vague and varies person to person.
Step 2: No fixed criterion is given.
Answer: No, it is not a set.

vi. All the letters of English Alphabet which precede K.

Step 1: The English alphabet is well-defined and ordered.
Step 2: Letters before K are: A, B, C, D, E, F, G, H, I, J.
Answer: Yes, it is a set.

vii. All the pet dogs in Meerut.

Step 1: We can clearly identify whether a dog is a pet and whether it resides in Meerut.
Step 2: Each pet dog in Meerut is unique.
Answer: Yes, it is a set.

viii. All the dishonest dealers in Delhi.

Step 1: The term “dishonest” is based on opinion and cannot be measured precisely.
Step 2: People may disagree on who is dishonest.
Answer: No, it is not a set.

ix. All the students of your school with age exceeding 12 years.

Step 1: This is measurable and specific (age > 12).
Step 2: Clear condition to include/exclude members.
Answer: Yes, it is a set.

x. All the girls of Meena’s class, who are taller than Meena.

Step 1: Height is a measurable quantity.
Step 2: Girls talller than Meena can be exactly identified.
Answer: Yes, it is a set.

Q2: Rewrite the following statements using set notation:

i. p is an element of set A.

Step 1: “is an element of” is denoted by the symbol ∈
Answer: \( p \in A \)

ii. q does not belong to set B.

Step 1: “does not belong to” is denoted by the symbol ∉
Answer: \( q \notin B \)

iii. a and b are members of set C.

Step 1: If both a and b belong to set C, we can write them using ∈.
Answer: \( a \in C,\ b \in C \)

iv. B and C are equivalent sets.

Step 1: Equivalent sets have the same number of elements.
Step 2: Denoted using the symbol ∼
Answer: \( B \sim C \)

v. Cardinal number of set E is 15.

Step 1: Cardinal number is represented as \( n(E) \)
Answer: \( n(E) = 15 \)

vi. A is an empty set and B is a non-empty set.

Step 1: Empty set is denoted by \( \emptyset \)
Step 2: B is non-empty means it contains at least one element.
Answer: \( A = \emptyset,\ B \ne \emptyset \)

vii. 0 is a whole number, but 0 is not a natural number.

Step 1: Whole numbers include 0: \( 0 \in W \)
Step 2: Natural numbers start from 1, so \( 0 \notin N \)
Answer: \( 0 \in W,\ 0 \notin N \)

Q3: Describe the following sets in roster form:

i. \( B = \{x \mid x \in W,\ x \le 6\} \)

Step 1: Whole numbers (W) start from 0.
Step 2: Take all values less than or equal to 6.
Answer: \( B = \{0, 1, 2, 3, 4, 5, 6\} \)

ii. \( C = \{x \mid x \text{ is a factor of } 32\} \)

Step 1: 32 is a even number.
Step 2: Its factors are 1, 2, 4, 8, 16 and 32.
Answer: \( C = \{1, 2, 4, 8, 16, 32\} \)

iii. \( E = \{x \mid x = 2n+1,\ n \in W,\ n \le 4\} \)

Step 1: Put values of \( n = 0, 1, 2, 3, 4 \)
Step 2: \( x = 1, 3, 5, 7, 9 \)
Answer: \( E = \{1, 3, 5, 7, 9\} \)

iv. \( F = \{x \mid x = n^2,\ n \in W,\ 2 \le n \le 5\} \)

Step 1: Take \( n = 2, 3, 4, 5 \)
Step 2: \( x = 4, 9, 16, 25 \)
Answer: \( F = \{4, 9, 16, 25\} \)

v. \( G = \left\{x \mid x = \frac{n}{n+3},\ n \in N,\ n \le 5\right\} \)

Step 1: Take \( n = 1, 2, 3, 4, 5 \)
Step 2: Calculate each value: \[ \frac{1}{4},\ \frac{2}{5},\ \frac{3}{6},\ \frac{4}{7},\ \frac{5}{8} \] Answer: \( G = \left\{ \frac{1}{4}, \frac{2}{5}, \frac{1}{2}, \frac{4}{7}, \frac{5}{8} \right\} \)

vi. \( H = \{x \mid x \text{ is a two-digit number, sum of whose digits is 8} \} \)

Step 1: Try all two-digit numbers where digits add to 8.
Step 2: Examples: 17, 26, 35, 44, 53, 62, 71, 80
Answer: \( H = \{17, 26, 35, 44, 53, 62, 71, 80\} \)

vii. \( I = \left\{x \mid x \in N,\ x \text{ is divisible both by 4 and 6},\ x \le 60\right\} \)

Step 1: Find LCM of 4 and 6 = 12
Step 2: Multiples of 12 ≤ 60: 12, 24, 36, 48, 60
Answer: \( I = \{12, 24, 36, 48, 60\} \)

viii. \( J = \left\{x \mid x = \frac{1}{n},\ n \in N,\ n \le 5 \right\} \)

Step 1: Take \( n = 1, 2, 3, 4, 5 \)
Step 2: \( x = \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \)
Answer: \( J = \left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\right\} \)

ix. \( L = \{x \mid x \text{ is a letter of the word ‘careless’} \} \)

Step 1: Extract all letters of the word: c, a, r, e, l, e, s, s
Step 2: Eliminate repetitions.
Answer: \( L = \{c, a, r, e, l, s\} \)

Q4: Describe the following sets in set-builder form:

i. A = {5,6,7,8,9,10,11,12}

Step 1: Identify the pattern – consecutive numbers from 5 to 12.
Answer: \( A = \{x \mid x \in \mathbb{N},\ 5 \le x \le 12\} \)

ii. B = {1,2,3,4,6,8,12,16,24,48}

Step 1: All elements are factors of 48.
Answer: \( B = \{x \mid x \text{ is a factor of } 48\} \)

iii. C = {11,13,17,19,23,29,31,37}

Step 1: All elements are prime numbers between 10 and 40.
Answer: \( C = \{x \mid x \text{ is a prime number},\ 10 < x < 40\} \)

iv. D = {21,23,25,27,29,31,33,35,37}

Step 1: All are odd numbers from 21 to 37.
Answer: \( D = \{x \mid x \text{ is odd},\ 21 \le x \le 37\} \)

v. I = {9,16,25,36,49,64,81,100}

Step 1: These are perfect squares of natural numbers.
Answer: \( I = \{x \mid x = n^2,\ n \in \mathbb{N},\ 3 \le n \le 10\} \)

vi. J = {-2, 2}

Step 1: Square of both numbers is 4.
Answer: \( J = \{x \mid x \in Z, x^2 = 4\} \)

vii. K = {0}

Step 1: Singleton set containing only zero.
Answer: \( K = \{x \mid x = 0\} \)

viii. L = { }

Step 1: This is the empty set.
Answer: \( L = \emptyset \ \text{or} \ \{x \mid x \ne x\} \)

ix. M = {a, b, c, d, e, f, g, h, i}

Step 1: These are first 9 letters of the English alphabet.
Answer: \( M = \{x \mid x \text{ is a letter of English alphabet},\ x \le i\} \)

x. \( P = \left\{ \frac{2}{7}, \frac{3}{8}, \frac{4}{9}, \frac{5}{10}, \frac{6}{11}, \frac{7}{12}, \frac{8}{13}, \frac{9}{14} \right\} \)

Step 1: Each element is of the form \( \frac{n}{n+5} \)
Step 2: Where \( n \in \mathbb{N},\ 2 \le n \le 9 \)
Answer: \( P = \left\{x \mid x = \frac{n}{n+5},\ n \in \mathbb{N},\ 2 \le n \le 9\right\} \)

xi. S = {Atlantic, Arctic}

Step 1: The set contains names of oceans.
Answer: \( S = \{x \mid x \text{ is an ocean and whose name begins with A} \} \)

xii. T = {Mars, Mercury}

Step 1: These are planets in the solar system.
Answer: \( T = \{x \mid x \text{ is a planet and whose name begins with M} \} \)

Q5: Separate finite and infinite sets from the following:

i. Set of leaves on a tree.

Step 1: A tree has a limited number of leaves, even if large.
Answer: Finite Set

ii. Set of all counting numbers.

Step 1: Counting numbers are infinite: 1, 2, 3, 4, …
Answer: Infinite Set

iii. \( \{x \mid x \in \mathbb{N},\ x > 1000\} \)

Step 1: Natural numbers greater than 1000 go on forever.
Answer: Infinite Set

iv. \( \{x \mid x \in \mathbb{W},\ x < 5000\} \)

Step 1: Whole numbers less than 5000 are from 0 to 4999.
Step 2: These are 5000 numbers in total.
Answer: Finite Set

v. \( \{x \mid x \in \mathbb{Z},\ x < 4\} \)

Step 1: Integers less than 4 include negative numbers and go infinitely.
Answer: Infinite Set

vi. Set of all triangles in a plane.

Step 1: A plane allows endless variations of triangle sizes and orientations.
Answer: Infinite Set

vii. Set of all points on the circumference of a circle.

Step 1: A circle contains infinite points along its circumference.
Answer: Infinite Set

viii. {1,2,3,1,2,3,1,2,3,…}

Step 1: The pattern keeps repeating endlessly but we cn easily count the repeating elements i.e. 1, 2, 3.
Answer: Finite Set

ix. \( \{x \mid x \in \mathbb{Q},\ 2 < x < 3\} \)

Step 1: There are infinitely many rational numbers between any two real numbers.
Answer: Infinite Set

Q6: Which of the following are empty sets?

i. A = \( \{x \mid x \in \mathbb{N},\ x+5=5\} \)

Step 1: Solve the equation: \( x + 5 = 5 \Rightarrow x = 0 \)
Step 2: But \( x \in \mathbb{N} \), and 0 ∉ Natural numbers.
Answer: A is an Empty Set

ii. B = \( \{x \mid x \in \mathbb{N},\ 2x+3=6\} \)

Step 1: Solve: \( 2x + 3 = 6 \Rightarrow x = \frac{3}{2} \)
Step 2: \( \frac{3}{2} \notin \mathbb{N} \)
Answer: B is an Empty Set

iii. C = \( \{x \mid x \in \mathbb{W},\ x + 2 < 2\} \)

Step 1: Solve: \( x + 2 < 2 \Rightarrow x < 0 \)
Step 2: But \( x \in \mathbb{W} \), i.e., \( x \ge 0 \)
Answer: C is an Empty Set

iv. D = \( \{x \mid x \in \mathbb{N},\ 1 < x \le 2\} \)

Step 1: Only value possible is \( x = 2 \)
Step 2: 2 ∈ \( \mathbb{N} \), and satisfies the condition
Answer: D is Not an Empty Set

v. E = \( \{x \mid x \in \mathbb{N},\ x^2 + 4 = 0\} \)

Step 1: \( x^2 + 4 = 0 \Rightarrow x^2 = -4 \)
Step 2: No real or natural number squared gives a negative result
Answer: E is an Empty Set

vi. F = \( \{x \mid x \text{ is a prime number},\ 90 < x < 96\} \)

Step 1: Check numbers between 91 and 95: 91, 92, 93, 94, 95
Step 2: None of these are prime
Answer: F is an Empty Set

vii. G = \( \{x \mid x \text{ is an even prime} \} \)

Step 1: 2 is the only even prime number
Answer: G is Not an Empty Set

viii. H = {0}

Step 1: The set contains the element 0
Answer: H is Not an Empty Set

Q7: Which of the following are pairs of equivalent sets?

i. A = {2, 3, 5, 7} and B = \( \{x \mid x \text{ is a whole number},\ x \le 4\} \)

Step 1: Set A has 4 elements: {2, 3, 5, 7}
Step 2: Set B includes whole numbers ≤ 4 → {0, 1, 2, 3, 4} → 5 elements
Answer: Not Equivalent

ii. C = \( \{x \mid x + 2 = 2\} \) and D = \( \phi \)

Step 1: Solve: \( x + 2 = 2 \Rightarrow x = 0 \)
Step 2: Since 0 is a valid solution, C = {0}, and D = empty set
Answer: Not Equivalent

iii. E = \( \{x \mid x \text{ is a natural number},\ x < 4 \} \) and F = \( \{x \mid x \text{ is a whole number},\ x < 3 \} \)

Step 1: E = {1, 2, 3} → 3 elements
Step 2: F = {0, 1, 2} → 3 elements
Answer: Equivalent Sets

iv. G = \( \{x \mid x \text{ is an integer},\ -3 < x < 3 \} \) and H = \( \{x \mid x \text{ is a factor of 16} \} \)

Step 1: G = { -2, -1, 0, 1, 2 } → 5 elements
Step 2: H = {1, 2, 4, 8, 16} → 5 elements
Answer: Equivalent Sets

Q8: State whether the following statements are true or false:

i. {a, b, c, 1, 2, 3} is not a set.

Step 1: This is a well-defined collection of distinct elements.
Step 2: Hence, it is a valid set.
Answer: False

ii. {5, 7, 9} = {9, 5, 7}

Step 1: Sets are unordered; order of elements doesn’t matter.
Answer: True

iii. \( \{x \mid x \in \mathbb{W},\ x + 8 = 8\} \) is a singleton set.

Step 1: Solve: \( x + 8 = 8 \Rightarrow x = 0 \), and 0 ∈ \( \mathbb{W} \)
Answer: True

iv. \( \{x \mid x \in \mathbb{W},\ x < 0\} = \phi \)

Step 1: Whole numbers start from 0, so no whole number is < 0.
Answer: True

v. \( \{x \mid x \in \mathbb{N},\ x + 5 = 3\} = \phi \)

Step 1: Solve: \( x = -2 \), and -2 ∉ \( \mathbb{N} \)
Answer: True

vi. \( \{x \mid x \in \mathbb{N},\ 3 < x \le 4\} = \phi \)

Step 1: x = 4 satisfies the condition and 4 ∈ \( \mathbb{N} \)
Answer: False

vii. If A = \( \{x \mid x \text{ is a letter of the word MEERAT} \} \), then n(A) = 6

Step 1: Unique letters in MEERAT = {M, E, R, A, T} → 5 letters
Answer: False

viii. If A = \( \{x \mid x \in \mathbb{N},\ 8 < x < 13\} \) and B = \( \{x \mid x \in \mathbb{Z},\ -3 \le x < 1\} \), then n(A) = n(B)

Step 1: A = {9, 10, 11, 12} → 4 elements
Step 2: B = {-3, -2, -1, 0} → 4 elements
Answer: True

ix. If n(A) = n(B), then A = B.

Step 1: Equal number of elements doesn’t mean the sets are equal.
Step 2: Equal sets must have same elements, not just same count.
Answer: False

x. If B is the set of all consonants in English alphabet, then n(B) = 21.

Step 1: English alphabet has 26 letters; 5 are vowels → 26 – 5 = 21 consonants
Answer: True

xi. \( \{x \mid x \text{ is a prime factor of 24} \} = \{2, 3, 4, 6, 8, 12, 24\} \)

Step 1: Prime factors of 24 are only {2, 3}
Step 2: The set on RHS includes non-prime and composite numbers
Answer: False