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Class 9 · Advanced MathematicsNCERT 2026 Revised Syllabus

Chapter 1: Sets

Class IX CBSE Advanced Mathematics

Introduces the language of sets — roster and set-builder forms, subsets and power sets, cardinality, union, intersection, difference and complement, with De Morgan\u2019s laws and applications of set theory.

Find complete NCERT solutions for Class 9 Advanced Mathematics, Class IX CBSE Advanced Mathematics Chapter 1. Every question has a clear, step-by-step answer.

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Exercise 1.1(13 questions)

1(a)

List the elements of the set $\{x : x \text{ is an integer and } x^2 = 9\}$.

School Level Solution

Sol.
Solution:Given: $x^2 = 9$.Taking square root on both sides: $$x = \pm 3$$Answer: $\boxed{\text{The set is }\{-3, 3\}}$.
1(b)

List the elements of the set $\{x : x \text{ is a positive integer less than } 5\}$.

School Level Solution

Sol.
Solution:Positive integers less than 5 are $1, 2, 3, 4$.Answer: $\boxed{\{1, 2, 3, 4\}}$.
1(c)

List the elements of the set $\{x : x \text{ is an even natural number divisible by } 5\}$.

School Level Solution

Sol.
Solution:A number that is both even and divisible by 5 must be a multiple of 10.Answer: $\boxed{\{10, 20, 30, \dots\}}$.
1(d)

List the elements of the set $\{x : x \in \mathbb{N} \text{ and } x < -1\}$.

School Level Solution

Sol.
Solution:Natural numbers are always positive. No natural number is negative, so no value of $x$ satisfies $x < -1$.Answer: $\boxed{\text{The set is the empty set }\varnothing}$.
2
Hard

Determine which elements of the set $\{-5, -\sqrt{3}, -\frac{1}{2}, 0, \frac{2}{5}, \pi, 13.4, \frac{1}{3}, \frac{19}{2}\}$ are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, (e) real numbers.

  1. a.Natural numbers
  2. b.Whole numbers
  3. c.Integers
  4. d.Rational numbers
  5. e.Real numbers

School Level Solution

Sol.
Solution:Compare each element of $A$ with the number system definitions.(a) Natural numbers — none of the elements is a natural number (the set contains no positive counting number). (b) Whole numbers — $0$ is the only whole number. (c) Integers — $-5$ and $0$. (d) Rational numbers — $-5$, $-\frac{3}{2}$, $-\frac{1}{5}$, $0$, $13.4$, $\frac{1}{3}$ (all can be written as a ratio of integers). $\sqrt{2}$, $\pi$, $\sqrt{19}$ are irrational. (e) Real numbers — all elements of $A$ are real numbers, so $A$ itself.Answer: $\boxed{\text{(a) None, (b) }\{0\}\text{, (c) }\{-5, 0\}\text{, (d) }\{-5, -\frac{3}{2}, -\frac{1}{5}, 0, 13.4, \frac{1}{3}\}\text{, (e) }A\text{ itself}}$.
3(a)

Write the set $\{x : x \text{ is a two-digit number and the sum of its digits is } 5\}$ in roster form.

School Level Solution

Sol.
Solution:Two-digit numbers whose digit sum is 5: $14 \ (1+4=5)$, $23 \ (2+3=5)$, $32 \ (3+2=5)$, $41 \ (4+1=5)$, $50 \ (5+0=5)$.Answer: $\boxed{\{14, 23, 32, 41, 50\}}$.
3(b)

Write the set $\{x : x \text{ is an integer and } |x| \geq 9\}$ in roster form.

School Level Solution

Sol.
Solution:$|x| \geq 9$ means $x \leq -9$ or $x \geq 9$. All such integers form an infinite set.Answer: $\boxed{\{\dots, -11, -10, -9, 9, 10, 11, \dots\}}$.
3(c)

Write the set $\{x : x \text{ is a letter of the word "SWEET"}\}$ in roster form.

School Level Solution

Sol.
Solution:The word "SWEET" has the letters S, W, E, E, T. A set lists each distinct element once.Answer: $\boxed{\{S, W, E, T\}}$.
3(d)
Hard

Write the set $\left\{x : x = \frac{n+1}{n},\ n \in \mathbb{N},\ n < 6\right\}$ in roster form.

School Level Solution

Sol.
Solution:For $n = 1, 2, 3, 4, 5$:$$x = \frac{n+1}{n}$$
$n$$x$
$1$$\frac{2}{1} = 2$
$2$$\frac{3}{2}$
$3$$\frac{4}{3}$
$4$$\frac{5}{4}$
$5$$\frac{6}{5}$
Answer: $\boxed{\left\{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \frac{6}{5}\right\}}$.
3(e)

Write the set $\{x : x \text{ is a composite number}\}$ in roster form.

School Level Solution

Sol.
Solution:Composite numbers are natural numbers greater than 1 that are not prime.Answer: $\boxed{\{4, 6, 8, 9, 10, 12, \dots\}}$.
4

Write the following sets in set-builder form: (i) $\{2, 4, 6, 8, \dots\}$, (ii) $\{3, 6, 9, 12, 15\}$, (iii) $\{1, 4, 9, 16, \dots\}$, (iv) $\{8, 9, 10, 11, \dots\}$, (v) $\{1, 2, 3, 6\}$.

  1. a.$\{2, 4, 6, 8, \dots\}$
  2. b.$\{3, 6, 9, 12, 15\}$
  3. c.$\{1, 4, 9, 16, \dots\}$
  4. d.$\{8, 9, 10, 11, \dots\}$
  5. e.$\{1, 2, 3, 6\}$

School Level Solution

Sol.
Solution:(i) $\{2, 4, 6, 8, \dots\}$ — even natural numbers: $$\{x : x = 2n, n \in \mathbb{N}\}$$(ii) $\{3, 6, 9, 12, 15\}$ — first five multiples of 3: $$\{x : x = 3n, n \in \mathbb{N}, n \leq 5\}$$(iii) $\{1, 4, 9, 16, \dots\}$ — perfect squares: $$\{x : x = n^2, n \in \mathbb{N}\}$$(iv) $\{8, 9, 10, 11, \dots\}$ — natural numbers from 8 onwards: $$\{x : x \in \mathbb{N}, x \geq 8\}$$(v) $\{1, 2, 3, 6\}$ — the factors of 6: $$\{x : x \text{ is a factor of } 6\}$$Note: Can two different sets have the same roster form? No. A roster form lists exactly the elements of one set, so two different sets cannot share the same roster form.Answer: $\boxed{\text{(i) }\{x : x = 2n, n \in \mathbb{N}\}\text{, (ii) }\{x : x = 3n, n \in \mathbb{N}, n \leq 5\}\text{, (iii) }\{x : x = n^2, n \in \mathbb{N}\}\text{, (iv) }\{x : x \in \mathbb{N}, x \geq 8\}\text{, (v) }\{x : x \text{ is a factor of } 6\}}$.
5
Hard

Which of the following pairs of sets are equal? (i) $\{D, E, C, E, N, T\}$ and $\{C, E, N, T, D\}$ (ii) $\{a, b, \pi, \sqrt{2}\}$ and $\{a, \pi, \sqrt{2}, b\}$ (iii) $\{x : x \text{ is a zero of the polynomial } x^2\}$ and $\{x : x \text{ is the root of the equation, } x^2 = 0\}$ (iv) $\{x : x \text{ has numerical value less than or equal to } 1\}$ and $\{x : x \text{ is the root of the equation, } x^2 - 1 = 0\}$ (v) $\{5, 10, 15, 20\}$ and $\{5, 10, 15, 20, \dots\}$ (vi) $\varnothing$ and $\{\varnothing\}$.

  1. a.$\{D, E, C, E, N, T\}$ and $\{C, E, N, T, D\}$
  2. b.$\{a, b, \pi, \sqrt{2}\}$ and $\{a, \pi, \sqrt{2}, b\}$
  3. c.$\{x : x \text{ is a zero of the polynomial } x^2\}$ and $\{x : x \text{ is the root of the equation, } x^2 = 0\}$
  4. d.$\{x : x \text{ has numerical value less than or equal to } 1\}$ and $\{x : x \text{ is the root of the equation, } x^2 - 1 = 0\}$
  5. e.$\{5, 10, 15, 20\}$ and $\{5, 10, 15, 20, \dots\}$
  6. f.$\varnothing$ and $\{\varnothing\}$

School Level Solution

Sol.
Solution:Two sets are equal if they contain exactly the same elements.(i) Same elements, only order/repetition differs. Equal. (ii) Same four elements; order does not matter. Equal. (iii) Both sets contain only $0$. Equal. (iv) The first set contains all real numbers with numerical value $leq 1$, while the second is ${-1,1}$. Not equal. (v) The first set is finite; the second is infinite. Not equal. (vi) $arnothing$ has 0 elements, while ${ arnothing}$ has 1 element. Not equal.Answer: $oxed{ ext{(i) Equal, (ii) Equal, (iii) Equal, (iv) Not equal, (v) Not equal, (vi) Not equal}}$.
6
Hard

State which of the following sets are finite or infinite: (i) $\{x : x \in \mathbb{Z} \text{ and } (x - 1)(x + 2)(x - 3) = 0\}$ (ii) $\{x : x \text{ and } 2 \text{ are coprime}\}$ (iii) $\{x : x \text{ is a rational number between } 3 \text{ and } 4\}$ (iv) $\{x : x \text{ is an integer and } |x| \geq 5\}$.

  1. a.$\{x : x \in \mathbb{Z} \text{ and } (x - 1)(x + 2)(x - 3) = 0\}$
  2. b.$\{x : x \text{ and } 2 \text{ are coprime}\}$
  3. c.$\{x : x \text{ is a rational number between } 3 \text{ and } 4\}$
  4. d.$\{x : x \text{ is an integer and } |x| \geq 5\}$

School Level Solution

Sol.
Solution:(i) $(x-1)(x+2)(x-3) = 0 \Rightarrow x = 1, -2, 3$. So the set is $\{1, -2, 3\}$. Finite.(ii) Numbers coprime to 2 are all odd numbers — infinitely many. Infinite.(iii) There are infinitely many rational numbers between 3 and 4. Infinite.(iv) $|x| \geq 5$ gives $x \leq -5$ or $x \geq 5$, so there are infinitely many integers. Infinite.Answer: $\boxed{\text{(i) Finite, (ii) Infinite, (iii) Infinite, (iv) Infinite}}$.

Exercise 1.2(5 questions)

1

Fill in the blanks with the symbols $\subset$ or $\not\subset$: (i) $\{2, 3, 4\} \_\_ \{1, 2, 3, 4, 5\}$ (ii) $\{x \mid x \text{ triangles in a plane}\} \_\_ \{x \mid x \text{ polygons in a plane}\}$ (iii) $\{x : x \text{ is an integer}\} \_\_ \{x : x \text{ is a multiple of } 4\}$ (iv) $\varnothing \_\_ \{\varnothing\}$ (v) $\{x \mid x = \frac{m-1}{m},\ m \text{ non-zero integer}\} \_\_ \{x \mid x \text{ is a rational number}\}$ (vi) $\{x \mid x = n^2\} \_\_ \{x \mid x = n^3\}$ (vii) $\{x \mid x \in \mathbb{R}\} \_\_ \{x \mid x = 2n\}$.

  1. a.$\{2, 3, 4\} \_\_ \{1, 2, 3, 4, 5\}$
  2. b.$\{x \mid x \text{ triangles}\} \_\_ \{x \mid x \text{ polygons}\}$
  3. c.$\{x : x \text{ integer}\} \_\_ \{x : x \text{ multiple of } 4\}$
  4. d.$\varnothing \_\_ \{\varnothing\}$
  5. e.$\{x \mid x = \frac{m-1}{m}\} \_\_ \{x \mid x \text{ rational}\}$
  6. f.$\{x \mid x = n^2\} \_\_ \{x \mid x = n^3\}$
  7. g.$\{x \mid x \in \mathbb{R}\} \_\_ \{x \mid x = 2n\}$

School Level Solution

Sol.
Solution:(i) Every element of $\{2, 3, 4\}$ is in $\{1, 2, 3, 4, 5\}$ → $\subset$. (ii) All triangles are polygons → $\subset$. (iii) Not every integer is a multiple of 4 (e.g. 3) → $\not\subset$. (iv) $\varnothing$ is a subset of every set → $\subset$. (v) $\frac{m-1}{m}$ is always rational for a non-zero integer $m$ → $\subset$. (vi) Not every square is a cube (e.g. 4 is not a perfect cube) → $\not\subset$. (vii) Not every real number is $2n$ (e.g. $\pi$) → $\not\subset$.Answer: $\boxed{\text{(i) }\subset\text{, (ii) }\subset\text{, (iii) }\not\subset\text{, (iv) }\subset\text{, (v) }\subset\text{, (vi) }\not\subset\text{, (vii) }\not\subset}$.
2

Determine whether the following statements are true or false: (i) $1 \in \{1\}$ (ii) $\{2\} \in \{2\}$ (iii) $\{2\} \in \{\{2\}\}$ (iv) $\varnothing \in \{1, 2, 3\}$ (v) $\varnothing \subset \{1, 2, 3\}$.

  1. a.$1 \in \{1\}$
  2. b.$\{2\} \in \{2\}$
  3. c.$\{2\} \in \{\{2\}\}$
  4. d.$\varnothing \in \{1, 2, 3\}$
  5. e.$\varnothing \subset \{1, 2, 3\}$

School Level Solution

Sol.
Solution:(i) $1 \in \{1\}$ — 1 is an element of the set. True. (ii) $\{2\} \in \{2\}$ — the set $\{2\}$ is not an element of itself. False. (iii) $\{2\} \in \{\{2\}\}$ — the set $\{2\}$ is an element of the outer set. True. (iv) $\varnothing \in \{1, 2, 3\}$ — empty set is not listed as an element. False. (v) $\varnothing \subset \{1, 2, 3\}$ — empty set is a subset of every set. True.Answer: $\boxed{\text{(i) True, (ii) False, (iii) True, (iv) False, (v) True}}$.
3

Write the power set of the following sets: (i) $\{1\}$ (ii) $\{p, q\}$ (iii) $\{1, 2, 5\}$ (iv) $\{\varnothing, \{\varnothing\}\}$.

  1. a.$\{1\}$
  2. b.$\{p, q\}$
  3. c.$\{1, 2, 5\}$
  4. d.$\{\varnothing, \{\varnothing\}\}$

School Level Solution

Sol.
Solution:The power set $P(A)$ contains all subsets of $A$.(i) $A = \{1\}$: $$P = \{\varnothing, \{1\}\}$$(ii) $A = \{p, q\}$: $$P = \{\varnothing, \{p\}, \{q\}, \{p, q\}\}$$(iii) $A = \{1, 2, 5\}$: $$P = \{\varnothing, \{1\}, \{2\}, \{5\}, \{1, 2\}, \{1, 5\}, \{2, 5\}, \{1, 2, 5\}\}$$(iv) $A = \{\varnothing, \{\varnothing\}\}$: $$P = \{\varnothing, \{\varnothing\}, \{\{\varnothing\}\}, \{\varnothing, \{\varnothing\}\}\}$$Answer: $\boxed{\text{(i) }P = \{\varnothing, \{1\}\}\text{, (ii) }P = \{\varnothing, \{p\}, \{q\}, \{p, q\}\}\text{, (iii) }P = \{\varnothing, \{1\}, \{2\}, \{5\}, \{1,2\}, \{1,5\}, \{2,5\}, \{1,2,5\}\}\text{, (iv) }P = \{\varnothing, \{\varnothing\}, \{\{\varnothing\}\}, \{\varnothing, \{\varnothing\}\}\}}$.
4

What is the cardinality of the following sets: (i) $\{a\}$ (ii) $\{a, \{a\}\}$ (iii) $\{\varnothing, 1, 2, \{1, 2\}\}$ (iv) $\{1, \{1\}, \{1, \{1\}\}\}$ (v) $\{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}$?

  1. a.$\{a\}$
  2. b.$\{a, \{a\}\}$
  3. c.$\{\varnothing, 1, 2, \{1, 2\}\}$
  4. d.$\{1, \{1\}, \{1, \{1\}\}\}$
  5. e.$\{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}$

School Level Solution

Sol.
Solution:Cardinality = number of distinct elements.(i) $\{a\}$ → $n = 1$ (ii) $\{a, \{a\}\}$ → $n = 2$ (iii) $\{\varnothing, 1, 2, \{1, 2\}\}$ → $n = 4$ (iv) $\{1, \{1\}, \{1, \{1\}\}\}$ → $n = 3$ (v) $\{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}$ → $n = 3$Answer: $\boxed{\text{(i) }n=1\text{, (ii) }n=2\text{, (iii) }n=4\text{, (iv) }n=3\text{, (v) }n=3}$.
5
Hard

Let $A$ be a set and $n(A) = 10$. Find $n[P(A)]$. What if $A$ has 100 elements?

School Level Solution

Sol.
Solution:The number of subsets of a set with $n$ elements is $2^n$: $$n[P(A)] = 2^{n(A)}$$For $n(A) = 10$: $$n[P(A)] = 2^{10} = 1024$$For $n(A) = 100$: $$n[P(A)] = 2^{100}$$Answer: $\boxed{2^{10} = 1024\text{ and }2^{100}\text{ respectively}}$.

Exercise 1.3(10 questions)

1

Find $A \cup B$: (i) $A = \{1, 2, 3, 7\}$, $B = \{2, 7, 9\}$ (ii) $A = \{a, b, d, e\}$, $B = \{a, e, i, o, u\}$ (iii) $A = \{x : x \in \mathbb{N}, x > 5\}$, $B = \{x : x \in \mathbb{N}, x < 5\}$ (iv) $A = \varnothing$, $B = \{2, 2, -1, 0\}$.

  1. a.$A = \{1, 2, 3, 7\}$, $B = \{2, 7, 9\}$
  2. b.$A = \{a, b, d, e\}$, $B = \{a, e, i, o, u\}$
  3. c.$A = \{x : x \in \mathbb{N}, x > 5\}$, $B = \{x : x \in \mathbb{N}, x < 5\}$
  4. d.$A = \varnothing$, $B = \{2, 2, -1, 0\}$

School Level Solution

Sol.
Solution:Union collects all elements from both sets.(i) $A \cup B = \{1, 2, 3, 7, 9\}$ (ii) $A \cup B = \{a, b, d, e, i, o, u\}$ (iii) $A$ has naturals greater than 5, $B$ has naturals less than 5: $$A \cup B = \mathbb{N} - \{5\}$$ (iv) Union with empty set leaves $B$ unchanged: $$A \cup B = \{2, 2, -1, 0\}$$Answer: $\boxed{\text{(i) }\{1, 2, 3, 7, 9\}\text{, (ii) }\{a, b, d, e, i, o, u\}\text{, (iii) }\mathbb{N} - \{5\}\text{, (iv) }\{2, 2, -1, 0\}}$.
2

Evaluate each of the following: (i) $\{1, 2\} \cap \{1, 2, 5\}$ (ii) $\{1, 3, 5, 7, 9\} \cap \{2, 4, 6, 8\} \cap \{a, t\}$ (iii) $\{g, o, a, t\} \cap \{c, a, t\}$ (iv) $\{x : x \text{ is an integer}\} \cap \{x : x \text{ is a negative integer}\}$.

  1. a.$\{1, 2\} \cap \{1, 2, 5\}$
  2. b.$\{1, 3, 5, 7, 9\} \cap \{2, 4, 6, 8\} \cap \{a, t\}$
  3. c.$\{g, o, a, t\} \cap \{c, a, t\}$
  4. d.$\{x : x \text{ integer}\} \cap \{x : x \text{ negative integer}\}$

School Level Solution

Sol.
Solution:Intersection keeps only the common elements.(i) $\{1, 2\} \cap \{1, 2, 5\} = \{1, 2\}$ (ii) No common element → $\varnothing$ (iii) $\{g, o, a, t\} \cap \{c, a, t\} = \{a, t\}$ (iv) Integers that are also negative integers are exactly the negative integers: $$\{x : x \text{ is a negative integer}\}$$Answer: $\boxed{\text{(i) }\{1, 2\}\text{, (ii) }\varnothing\text{, (iii) }\{a, t\}\text{, (iv) }\{x : x \text{ is a negative integer}\}}$.
3

Which of the following sets are disjoint? (i) $\{x : \text{multiple of } 2\}$ and $\{x : \text{multiple of } 3\}$ (ii) $\{e, \pi, \sqrt{2}, 0\}$ and $\{e^2, \sqrt{\pi}, 3, 1\}$ (iii) $\{x : x \text{ real number}\}$ and $\{x : x \text{ irrational number}\}$.

  1. a.$\{x : \text{multiple of } 2\}$ and $\{x : \text{multiple of } 3\}$
  2. b.$\{e, \pi, \sqrt{2}, 0\}$ and $\{e^2, \sqrt{\pi}, 3, 1\}$
  3. c.$\{x : x \text{ real}\}$ and $\{x : x \text{ irrational}\}$

School Level Solution

Sol.
Solution:(i) Multiples of 2 and multiples of 3 both contain 6. Not disjoint. (ii) $\{e, \pi, \sqrt{2}, 0\}$ and $\{e^2, \sqrt{\pi}, 3, 1\}$ share no common element. Disjoint. (iii) Every irrational number is real, so they overlap. Not disjoint.Answer: $\boxed{\text{(i) Not disjoint, (ii) Disjoint, (iii) Not disjoint}}$.
4

Find $A - B$ in each of the following: (i) $A = \{1, 3, 5, 8\}$, $B = \{3, 7, 8, 9\}$ (ii) $A = \{3, 0, 8\}$, $B = \{1, 3, 0, 8, 9\}$ (iii) $A = \{2, 6\}$, $B = \{1, 3, 5, 9\}$.

  1. a.$A = \{1, 3, 5, 8\}$, $B = \{3, 7, 8, 9\}$
  2. b.$A = \{3, 0, 8\}$, $B = \{1, 3, 0, 8, 9\}$
  3. c.$A = \{2, 6\}$, $B = \{1, 3, 5, 9\}$

School Level Solution

Sol.
Solution:$A - B$ = elements of $A$ not in $B$.(i) $A - B = \{1, 3, 5, 8\} - \{3, 7, 8, 9\} = \{1, 5\}$ (ii) Every element of $A$ is in $B$, so $A - B = \varnothing$ (iii) $A - B = \{2, 6\}$ (nothing from $A$ is in $B$)Answer: $\boxed{\text{(i) }\{1, 5\}\text{, (ii) }\varnothing\text{, (iii) }\{2, 6\}}$.
5
Hard

Using the Venn diagram with $A = \{1, 11, 4, 8, 5\}$, $B = \{4, 2, 5, 3\}$, $C = \{8, 5, 3, 6, 9\}$, $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13\}$, find: (i) $A\text{\u2019}$ (ii) $B\text{\u2019}$ (iii) $(A \cap B)\text{\u2019}$ (iv) $A\text{\u2019} \cup B\text{\u2019}$ (v) $A \cap B \cap C$ (vi) $A \cap (B \cup C)$.

  1. a.$A\text{\u2019}$
  2. b.$B\text{\u2019}$
  3. c.$(A \cap B)\text{\u2019}$
  4. d.$A\text{\u2019} \cup B\text{\u2019}$
  5. e.$A \cap B \cap C$
  6. f.$A \cap (B \cup C)$

School Level Solution

Sol.
Solution:Complement $= U - \text{set}$.(i) $A\text{\u2019} = U - A = \{2, 3, 6, 7, 9, 10, 13\}$ (ii) $B\text{\u2019} = U - B = \{1, 6, 7, 8, 9, 10, 11, 13\}$ (iii) $A \cap B = \{4, 5\}$, so $(A \cap B)\text{\u2019} = \{1, 2, 3, 6, 7, 8, 9, 10, 11, 13\}$ (iv) By De Morgan\u2019s Law: $A\text{\u2019} \cup B\text{\u2019} = (A \cap B)\text{\u2019} = \{1, 2, 3, 6, 7, 8, 9, 10, 11, 13\}$ (v) $A \cap B \cap C = \{5\}$ (vi) $B \cup C = \{2, 3, 4, 5, 6, 8, 9\}$, so $A \cap (B \cup C) = \{4, 5, 8\}$Answer: $\boxed{\text{(i) }\{2, 3, 6, 7, 9, 10, 13\}\text{, (ii) }\{1, 6, 7, 8, 9, 10, 11, 13\}\text{, (iii) }\{1, 2, 3, 6, 7, 8, 9, 10, 11, 13\}\text{, (iv) }\{1, 2, 3, 6, 7, 8, 9, 10, 11, 13\}\text{, (v) }\{5\}\text{, (vi) }\{4, 5, 8\}}$.
6
Hard

Verify $A - B = A \cap B\text{\u2019}$ using the given Venn diagram: $A = \{1, 8, 5, 13, 6\}$, $B = \{2, 3, 5, 13, 7\}$, $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 13\}$.

School Level Solution

Sol.
Solution:Step 1: $A - B = \{1, 8, 6\}$ (remove elements of $B$ from $A$).Step 2: $B\text{\u2019} = U - B = \{1, 4, 6, 8, 9\}$.Step 3: $A \cap B\text{\u2019} = \{1, 6, 8\} = A - B$.Verified: $A - B = A \cap B\text{\u2019}$.Answer: $\boxed{\text{Verified: }A - B = A \cap B\text{\u2019} = \{1, 6, 8\}}$.
7
Hard

In a class, 85% opted Mathematics and 75% opted Science. (a) Find the minimum percentage who opted for both. (b) If 10% opted neither subject, what is the minimum percentage who opted for both?

  1. a.Minimum % who opted for both
  2. b.If 10% opted neither, find the minimum % for both

School Level Solution

Sol.
Solution:(a) Minimum who opted for both: $$85 + 75 - 100 = 60\%$$(b) If 10% opted neither, those with at least one $= 90\%$: $$\text{Minimum both} = 85 + 75 - 90 = 70\%$$Answer: $\boxed{\text{(a) 60%, (b) 70%}}$.
8
Hard

In a survey of 50 people, 35 speak English and 25 speak Hindi, and every person speaks at least one of the two. Find the number $k$ who speak only English.

School Level Solution

Sol.
Solution:Let $S$ = all 50 people, $E$ = English speakers, $H$ = Hindi speakers.$$n(E \cup H) = 50$$ $$n(E \cap H) = 35 + 25 - 50 = 10$$Only English speakers: $$k = 35 - 10 = 25$$Answer: $\boxed{k = 25}$.
9
Hard

In a group of 56 students, 17 did Origami, 28 played an Instrument, 25 did Fine arts, 4 did all three, and every student got at least one certificate. How many students got certificates in exactly two activities?

School Level Solution

Sol.
Solution:$$56 = 17 + 28 + 25 - [\text{pairwise sum}] + 4$$ $$56 = 74 - [\text{pairwise sum}]$$ $$\text{pairwise sum} = 18$$Exactly two $=$ pairwise sum $- 3 \times$ (all three) $= 18 - 12 = 6$.Answer: $\boxed{\text{6 students}}$.
10
Hard

In a group of 100 students: English 60, German 50, Spanish 35; English and German 40, German and Spanish 30, English and Spanish 25; all three 25. Find (a) students knowing at least two languages, (b) students knowing at most one language, (c) students knowing none.

  1. a.At least two languages
  2. b.At most one language
  3. c.None

School Level Solution

Sol.
Solution:Let $E, G, S$ denote the three language groups with $n(U) = 100$.(a) At least two $= (E\cap G + G\cap S + E\cap S) - 2(E\cap G\cap S)$ $$= 40 + 30 + 25 - 50 = 45$$(b) At least one $= E + G + S - (E\cap G + G\cap S + E\cap S) + E\cap G\cap S$ $$= 60 + 50 + 35 - 95 + 25 = 75$$ $$\text{At most one} = 100 - 45 = 55$$(c) None $= 100 - 75 = 25$.Answer: $\boxed{\text{(a) 45, (b) 55, (c) 25 students}}$.

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