Math HW

Set Theory: DPPs & Homework Questions

---

Lecture 1

• Homework Assigned: Complete your notes before the next class and revise them.
• DPP: No DPP was assigned for this introductory class.

---

Lecture 2

• Homework Assigned: Re-attempt all class examples, solve the DPP, and complete NCERT Exercise 1.1.

---

Lecture 3

• Homework Assigned: Re-attempt all class examples, complete the DPP, and solve NCERT Exercise 1.1.

DPP-01

Q1. Which of the following is an example of a null set?
(A) Set of odd natural numbers divisible by 2
(B) Set of even prime numbers
(C) $\{x:x\text{ is a natural number, }x>5\text{ and }x<7\}$
(D) $\{y:y\text{ is a point common to any two parallel lines}\}$

Q2. If A is the set of days in a week, then set A is a:
(A) Empty set
(B) Singleton set
(C) Finite set
(D) Infinite set

Q3. If $P=\{x:x\text{ is a prime number less than 20}\}$ and $M=\{x:x\text{ is a multiple of 6, }0<x<30\}$, then $n(P)-n(M)$ is:
(A) 2
(B) 4
(C) 5
(D) 8

Q4. Which of the following collections is well defined?
(A) Collection of most famous movies.
(B) Collection of intelligent students.
(C) Collection of delicious food.
(D) Collection of students scored more than 50 marks in Maths test.

Q5. Correct set-builder form of set $A=\{-2,-1,0,1,2,3\}$ is:
(A) $A=\{x\mid -3<x<4,x\in R\}$
(B) $A=\{x\mid -3<x<4,x\in N\}$
(C) $A=\{x\mid -3<x<4,x\in Z\}$
(D) $A=\{x\mid -3<x<4,x\in W\}$

Q6. Correct tabular form of set $X=\{x\mid x\text{ is letter of word SUCCESS, x is a vowel}\}$ is:
(A) $X=\{S,U,C,E\}$
(B) $X=\{S,S,S,U,E,C,C\}$
(C) $X=\{E,U\}$
(D) $X=\{U,E,C\}$

Q7. If set X is collection of all the prime numbers less than 40 then $n(X)=$
(A) 39
(B) 12
(C) 13
(D) 11

Q8. Which of following statement is incorrect?
(A) $R=Q+\bar{Q}$
(B) $I^{+}=N$
(C) $0\in R^{+}$
(D) $R^{+}=(0,\infty )$

Q9. The set $A=\{x:x\in R,x^2=16\text{ and }2x=6\}$ in roaster form is:
(A) $\varphi$
(B) $\{14,3,4\}$
(C) $\{3\}$
(D) $\{4\}$

Q10. If $Q=\{x:x^2-x-6=0\text{ and }x\in N\}$ then the set in roaster form is:
(A) $\{3\}$
(B) $\{2\}$
(C) $\{3,-2\}$
(D) $\varphi$

---

Lecture 4

• Homework Assigned: Re-attempt all class examples, solve the DPP, and complete NCERT Exercise 1.1.

DPP-02

Q1. Write the set $\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7}\}$ in the set-builder form:
(A) $\{x:x=\frac{n}{n+1}\text{ where }n\text{ is a natural number and }1\le n\le 6\}$
(B) $\{x:x=\frac{n}{n+1}\text{ where }1\le n\le 6\}$
(C) $\{x:x=\frac{n}{n+1}\text{ where }n\text{ is a integer}\}$
(D) $\{x:x=\frac{n}{n+1}\text{ where }n\text{ is a whole number}\}$

Q2. The set $A=\{1,4,9,16,25,\dots \}$ in set-builder form is:
(A) $\{x:x=n^2,\text{ where }n\in N\}$
(B) $\{x:x=n^2,\text{ where }n\in Z\}$
(C) $\{x:x=2n+1,\text{ where }n\in Z\}$
(D) $\{x:x=(2n+1{)}^2,\text{ where }n\in N\}$

Q3. The set of the counting numbers which are multiples of 6 and less than 50 is written in set builder form as:
(A) $\{x:x\in N\text{ and is a multiple of 6}\}$
(B) $\{x:x=6\text{ and }0<x<50\}$
(C) $\{x:x\text{ is a multiple of 6 and }0<x<50,x\in N\}$
(D) $\{x:x\text{ is a multiple of 6 and }x\le 50\}$

Q4. If $B=\{x:x=\frac{2n-1}{n+1},n\in W\text{ and }n<4\}$, then B can be written as:
(A) $\{6,\frac{1}{3},\frac{2}{3},-\frac{1}{2}\}$
(B) $\{-\frac{1}{2},\frac{1}{3},\frac{3}{4},1\}$
(C) $\{3,-\frac{1}{2},\frac{3}{4},\frac{6}{5}\}$
(D) $\{-1,\frac{1}{2},1,\frac{5}{4}\}$

Q5. Which of following is equal to set $A=\{x\mid x^2+2x+24>0,x\in R\}$?
(A) $B=\{6,4\}$
(B) $R$
(C) $B=\{-6,4\}$
(D) $\{\}$

Q6. $A=\{x:x\ne x\}$ represents:
(A) $\{0\}$
(B) $\{\}$
(C) $\{1\}$
(D) $\{x\}$
kmsdfk
Q7. Which of the following is a null set?
(A) $A=\{x:x>1\text{ and }x<1\}$
(B) $B=\{x:x+3=3\}$
(C) $C=\{\varphi \}$
(D) $D=\{x:x\ge 1\text{ and }x\le 1\}$

Q8. Which of the following is an empty set?
(A) $\{x:x\text{ is a real number and }{x}^2+2=0\}$
(B) $\{x:x\text{ is a real number and }{x}^2-2=0\}$
(C) $\{x:x\text{ is a real number and }(x-1)(x-2)(x-3)=0\}$

---

Lecture 5

• Homework Assigned: Re-attempt all Class Examples, DPP, NCERT-Exercise-1.1

DIBY Questions (01 to 04)

DIBY-01. State whether the following collections is a set or not?
(i) The collection of natural numbers between 2 and 20
(ii) The collection of numbers which satisfy the equation $x^2-5x+6=0$
(iii) The collection of prime numbers between 1 and 100.
(iv) The collection of all intelligent women in Jalandhar

DIBY-02. Write the following set in builder form:
(i) Set of all rational number
(ii) $\{2,5,10,17,26,37,\dots \}$

DIBY-03. Consider set $A=\{1,2,3,4,5,6,7\}$ then find:
(i) number of subsets of set A
(ii) number of proper subsets of set A
(iii) number of non-empty subsets of set A.
(iv) number of non-empty proper subsets of set A.
(v) number of non-empty subsets containing only odd numbers.
(vi) number of subsets of A whose product of elements is even.
(vii) number of subsets of A whose product of elements is odd.

DIBY-04. Two finite sets have m and n elements respectively. The total number of subsets of first set is 992 more than the total number of subsets of the second set. Find the value of m & n.

DPP-03

Q1. Find the pairs of equal sets, if any: $A=\{0\}$, $B=\{x:x>15\text{ and }x<5\}$, $C=\{x:x-5=0\}$, $D=\{x:x^2=25\}$, $E=\{x:x\text{ is an integral positive root of }{x}^2-2x-15=0\}$
(A) A and B
(B) D and E
(C) C and D
(D) C and E

Q2. Which of the following represents a pair of equal sets?
(A) $A=\{1,2,3\}$, $B=\{x:x\in N\text{ and }x>5\}$
(B) $A=\{B,O,W,L\}$, $B=\{x:x\text{ is a letter of the word ELBOW}\}$
(C) $A=\{1,4,9,16\}$, $B=\{x:x=n^2,n\in N\text{ and }0<n<5\}$
(D) $A=\varphi$, $B=\{O,E,A\}$

Q3. If $A=\{1,2,3,4,5\}$, then the number of proper subsets of A is:
(A) 120
(B) 30
(C) 31
(D) 32

Q4. Which of following set is equivalent to set $X=\{x\mid x^2-11x+30=0,x\in R\}$?
(A) Set of all the letters of word SMALL
(B) Set of all even prime numbers
(C) Set of all integers greater than 4 less than 8
(D) Set of all non-zero integer whose modulus is less than 2.

Q5. If $n(A)=2$, then $n(P(P(A)))=$
(A) 16
(B) $2^{16}$
(C) 2
(D) $2^4$

Q6. Which set is the subset of all given sets?
(A) $\{1,2,3,4\dots \}$
(B) $\{1\}$
(C) $\{0\}$
(D) $\{\}$

Q7. If set A has n elements, then the total number of subsets of A is:
(A) $n$
(B) $n^2$
(C) $2^n$
(D) $2n$

Q8. If $A=\{x,y\}$, then the power set of A is:
(A) $\{x^x,y^y\}$
(B) $\{\varphi ,x,y\}$
(C) $\{\varphi ,\{x\},\{2y\}\}$
(D) $\{\varphi ,\{x\},\{y\},\{x,y\}\}$

Q9. Which of the following is a true statement:
(A) $\{a\}\in \{a,b,c\}$
(B) $\{a\}\subseteq \{a,b,c\}$
(C) $\varphi \in \{a,b,c\}$
(D) None of these

Q10. For a set $\{1,2,\{1,2,3\}\}$. Which of the following statement is false?
(A) $\{1,2\}\in \{1,2,\{1,2,3\}\}$
(B) $\{1,2\}\subseteq \{1,2,3\}$
(C) $\{1,2,3\}\subseteq \{1,2,\{1,2,3\}\}$
(D) $2\in \{1,2,\{1,2,3\}\}$

---

Lecture 6

• Homework Assigned: Re-attempt all Class Examples, DPP, NCERT-Exercise-1.1

DIBY Questions (05 to 10)

DIBY-05. Let $A=\{3,6,9,12,15,18,21\}$, $B=\{4,8,12,16,20\}$, $C=\{2,4,6,8,10,12,14,16\}$ and $D=\{5,10,15,20\}$. Which of the following is correct?
I. $(A\cup B)-C=\{3,9,18,21\}$
II. $(A-B)\cap (C-D)=\{6\}$
III. $(A\cap B\cap C)\cup D=\{12,5,15,10,20\}$
(A) Only I and II
(B) Only II and III
(C) Only III and I
(D) None of these

DIBY-06. If A and B are two sets such that $n(A)=70$, $n(B)=60$, and $n(A\cup B)=110$, then $n(A\cap B)$ is equal to:
(A) 240
(B) 50
(C) 40
(D) 20

DIBY-07. Let $n(U)=700$, $n(A)=200$, $n(B)=300$ and $n(A\cap B)=100$ then $n(A^{\prime }\cap B^{\prime })$ is:
(A) 400
(B) 600
(C) 300
(D) 200

DIBY-08. A market research group conducted a survey of 1000 consumers and reported that 720 consumers liked product A and 450 consumers liked product B. What is the least number that must have liked both products?
(A) 170
(B) 280
(C) 220
(D) None of these

DIBY-09. If A, B and C are any three sets, then $A-(B\cup C)$ is equal to:
(A) $(A-B)\cup (A-C)$
(B) $(A-B)\cap (A-C)$
(C) $(A-B)\cup C$
(D) $(A-B)\cap C$

DIBY-10. If A, B and C are non-empty sets, then $(A-B)\cup (B-A)$ equals:
(A) $(A\cup B)-B$
(B) $A-(A\cap B)$
(C) $(A\cup B)-(A\cap B)$
(D) $(A\cap B)\cup (A\cup B)$

---

Lecture 7

• Homework Assigned: Re-attempt all Class Examples, DPP, NCERT-COMPLETE, 2-sitting!

DIBY Questions (11 to 20)

DIBY-11. What does the shaded portion of the 3-circle Venn diagram (representing exactly two sets overlapping) represent?
(A) $(P\cap Q)\cap (P\cap R)$
(B) $((P\cap Q)-R)\cup ((P\cap R)-Q)$
(C) $((P\cup Q)-R)\cap ((P\cap R)-Q)$
(D) $((P\cap Q)\cup R)\cap ((P\cup R)-R)$

DIBY-12. In a class of 55 students, the number of students studying different subjects are 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is:
(A) 6
(B) 9
(C) 7
(D) 22

DIBY-13. If $X=\{1,2,3,4,5,6,7,8,9\}$, $A=\{x\in N:30<x^2<70\}$, $B=\{x:x\text{ is a prime number less than 10}\}$, then which of the following is incorrect?
(A) $A\cup B=\{2,3,5,6,7,8\}$
(B) $A\cap B=\{7,8\}$
(C) $A-B=\{6,8\}$
(D) $A\Delta B=\{2,3,5,6,8\}$

DIBY-14. If A and B be two universal sets and $A\cup B\cup C=U$. Then, $((A-B)\cup (B-C)\cup (C-A){)}^{\prime }$ is equal to:
(A) $A\cup B\cup C$
(B) $A\cup (B\cap C)$
(C) $A\cap B\cap C$
(D) $A\cap (B\cup C)$

DIBY-15. If there are three athletic teams in a school 21 are in the basketball team, 26 in hockey team and 29 in the football team. 14 play hockey and basketball, 15 play hockey and football, 12 play football and basketball and 8 play all the games. Then total number of members are:
(A) 42
(B) 43
(C) 45
(D) None of these

DIBY-16. Let A, B, and C be three sets such that $n(A)=40,n(B)=20,n(C)=30,n(A\cap B)=8,n(B\cap C)=10,n(C\cap A)=12$, and $n(A\cap B\cap C)=5$. Find the number of elements in exactly one of the three sets.
(A) 35
(B) 40
(C) 45
(D) 50

DIBY-17. Consider the sets $A=\{n\in N\mid n\text{ is a factor of 36}\}$ and $B=\{n\in N\mid n\text{ is a factor of 48}\}$. Find $n(A\Delta B)$.
(A) 4
(B) 6
(C) 7
(D) 9

DIBY-18. In a class of 50 students, 30 like coffee, 25 like tea, and 15 like both. How many students like neither coffee nor tea?
(A) 5
(B) 10
(C) 15
(D) 20

DIBY-19. If $P(A)$ denotes the power set of A, and $A=\{1,2,\{3,4\}\}$, find $n(P(A))$.
(A) 4
(B) 8
(C) 16
(D) 32

DIBY-20. Let $X=\{1,2,3,\dots ,10\}$ and $A=\{1,2,3,4,5\}$. How many subsets B of X exist such that $B\cap A=\varphi$?
(A) 16
(B) 32
(C) 64
(D) 128

---

Lecture 8

• Homework Assigned: Re-attempt all Class Examples, DPP-03 (Q 4-7), DIBY (7, 12, 16), NCERT-COMPLETE, Short Notes.

DPP-04

Q8. If A, B are two sets such that $A\cap B=B$ then:
(A) $B\subset A$
(B) $A\subset B$
(C) $A\cup B=B$
(D) None of these

Q9. If A, B, C are three sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C$ then:
(A) $A=B$
(B) $B=C$
(C) $A=C$
(D) $A=B=C$

DIBY Questions (21 to 31)

DIBY-21. Given $n(U)=100,n(A)=40,n(B)=50,n(A\cap B)=15$. Match the quantities in Column-I with their values in Column-II.
Column-I: [A. $n(A\cup B)$], [B. $n(A^{\prime })$], [C. $n(A\setminus B)$], [D. $n(B\setminus A)$]
Column-II: [p. 25], [q. 35], [r. 60], [s. 75]
(A) A $\to$ p, B $\to$ q, C $\to$ r, D $\to$ s
(B) A $\to$ p, B $\to$ r, C $\to$ q, D $\to$ s
(C) A $\to$ s, B $\to$ r, C $\to$ p, D $\to$ q
(D) A $\to$ s, B $\to$ p, C $\to$ r, D $\to$ q

DIBY-22. Let A, B, and C be sets such that $n(A)=10$, $n(B)=15$, $n(C)=20$, $n(A\cap B)=4$, $n(B\cap C)=6$, $n(A\cap C)=5$, and $n(A\cup B\cup C)=32$. Find $n(A\cap B\cap C)$.

DIBY-23. Given the sets $A=\{1,2,3\}$, $B=\{3,4\}$, $C=\{4,5,6\}$, then $A\cup (B\cap C)$ is:
(A) $\{3\}$
(B) $\{1,2,3,4\}$
(C) $\{1,2,4,5\}$
(D) $\{1,2,3,4,5,6\}$

DIBY-24. There are 200 individuals with skin disorder, out of them 100 have been exposed to Chemical $C_1$, 50 to Chemical $C_2$, and 30 to both $C_1$ & $C_2$. Find the number of individuals exposed to $C_1$ but not $C_2$.

DIBY-25. Evaluate the unknowns in each of the following cases:
(a) $n(A\cup B),n(A\cap B)$, if $A\subset B,n(A)=30,n(B)=36$
(b) $n(B)$, if $n(A)=30,n(A\cup B)=50,n(A\cap B)=10$
(c) $n(A\cup B)$, if $n(A)=20,n(A^{\prime }\cap B)=8$
(d) $n(A\cap B)$, if $n(A-B)=30,n(B-A)=16,n(A\cup B)=60$
(e) $n(A\cup B)$, if $n(S)=70,n(A)=40,n(B)=50,n(A\cap \overline{B})=27$

DIBY-26. A survey shows that 61%, 46% and 29% of the people watched “3 idiots”, “Rajneeti” and “Avatar” respectively. 25% people watched exactly two of the three movies and 3% watched none. What percentage of people watched all the three movies?

DIBY-27. The set $(A\cap B^{\prime }{)}^{\prime }\cup (B\cap C)$ is equal to:
(A) $A^{\prime }\cup B\cup C$
(B) $A^{\prime }\cup B$
(C) $A^{\prime }\cap C^{\prime }$
(D) $A^{\prime }\cap B$

DIBY-28. Let X be the universal set for sets A and B. If $n(A)=200$, $n(B)=300$ and $n(A\cap B)=100$, then $n(A^{\prime }\cap B^{\prime })$ is equal to 300 provided $n(X)$ is equal to:
(A) 600
(B) 700
(C) 800
(D) 900

DIBY-29. In a group of 50 people, two tests were conducted, one for diabetes and one for blood pressure. 30 people were diagnosed with diabetes and 40 people were diagnosed with high blood pressure. What is the minimum number of people who were having both diabetes and high blood pressure?
(A) 0
(B) 10
(C) 20
(D) 30

DIBY-30. A class has 175 students. The following data shows the number of students obtaining one or more subjects. Mathematics 100, Physics 70, Chemistry 40; Mathematics and Physics 30, Mathematics and Chemistry 28, Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students have offered Mathematics alone?
(A) 35
(B) 48
(C) 60
(D) 22

DIBY-31. In a class of 80 students numbered 1 to 80, all odd numbered students opt for Cricket, students whose numbers are divisible by 5 opt for Football and those whose numbers are divisible by 7 opt for Hockey. If the number of students who do not opt any of the three games is n, then $\frac{n}{4}$ is equal to?

---

---

Master Answer Key

DPP Answers

DPP-01
Q1: (A) & (D)
Q2: (C)
Q3: (B)
Q4: (D)
Q5: (C)
Q6: (C)
Q7: (B)
Q8: (C)
Q9: (A)
Q10: (A)

DPP-02
Q1: (A)
Q2: (A)
Q3: (C)
Q4: (D)
Q5: (B)
Q6: (B)
Q7: (A)
Q8: (A)

DPP-03
Q1: (D)
Q2: (C)
Q3: (C)
Q4: (D)
Q5: (A)
Q6: (D)
Q7: (C)
Q8: (D)
Q9: (B)
Q10: (C)

DPP-04
Q8: (A)
Q9: (B)

---

DIBY Answers

DIBY-01: (i) Yes, (ii) Yes, (iii) Yes, (iv) No
DIBY-02: (i) $\{x:x=\frac{p}{q},p\in I,q\in N\}$ (ii) $\{x:x=n^2+1,n\in N\}$
DIBY-03: (i) 128 (ii) 127 (iii) 127 (iv) 126 (v) 15 (vi) 112 (vii) 15
DIBY-04: $m=10$, $n=5$
DIBY-05: (B)
DIBY-06: (D)
DIBY-07: (C)
DIBY-08: (A)
DIBY-09: (B)
DIBY-10: (C)
DIBY-11: (B)
DIBY-12: (D)
DIBY-13: (D)
DIBY-14: (D)
DIBY-15: (B)
DIBY-16: (C)
DIBY-17: (C)
DIBY-18: (B)
DIBY-19: (B)
DIBY-20: (B)
DIBY-21: (C)
DIBY-22: 2
DIBY-23: (B)
DIBY-24: 70
DIBY-25: (a) $n(A\cup B)=36$; $n(A\cap B)=30$, (b) $n(B)=30$, (c) $n(A\cup B)=28$, (d) $n(A\cap B)=14$, (e) $n(A\cup B)=140$
DIBY-26: 7%
DIBY-27: (B)
DIBY-28: (B)
DIBY-29: (C)
DIBY-30: (C)
DIBY-31: 7