Which of the following are sets? Justify your answer.

The collection of all whole numbers less than 10.

Which of the following are sets? Justify your answer.

The collection of good hockey players in India.

Which of the following are sets? Justify your answer.

The collection of all the questions in this chapter.

Which of the following are sets? Justify your answer.

The collection of all the difficult chapters in this book.

Which of the following are sets? Justify your answer.

A collection of Hindi novels written by Munshi Prem Chand.

Which of the following are sets? Justify your answer.

A team of 11 best cricket players of India.

Which of the following are sets? Justify your answer.

The collection of all the months of the year whose names begin with the letter M.

Which of the following are sets? Justify your answer.

The collection of all interesting books.

Which of the following are sets? Justify your answer.

The collection of all short boys of your class.

Which of the following are sets? Justify your answer.

The collection of all those students of your class whose ages exceed 15 years.

Which of the following are sets? Justify your answer.

The collection of all rich persons of Kolkata.

Which of the following are sets? Justify your answer.

The collection of all persons of Kolkata whose assessed annual incomes exceed (say) Rs 20 lakh in the 4 financial years 2016-17.

Which of the following are sets? Justify your answer.

The collection of all interesting dramas written by Shakespeare.

Let A be the set of all even whole numbers less than 10.

(i) Write A in the roster from.

(ii) Fill in the blanks with the approximate symbol ∉ or ϵ :

(a) 0 …. A

(b) 10 …. A

(c) 3 …. A

(d) 6 …. A

Write the following sets in roster from:

A = {x : x is a natural number, 30 ≤ x < 36}.

Write the following sets in roster from:

B = {x : x is an integer and –4 < x < 6}.

Write the following sets in roster from:

C = {x : x is a two-digit number such that the sum of its digits is 9}.

Write the following sets in roster from:

D = {x : x is an integer, x² ≤ 9}.

Write the following sets in roster from:

E = {x : x is a prime number, which is a divisor of 42}.

Write the following sets in roster from:

F = {x : x is a letter in the word’ MATHEMATICS’}.

Write the following sets in roster from:

G = {x : x is a prime number and 80 < x < 100}.

Write the following sets in roster from:

H = {x : x is a perfect square and x < 50}.

Write the following sets in roster from:

J = {x : ϵ R and x² + x – 12 = 0}.

Write the following sets iroster from:

K = {x : ϵ N, x is a multiple of 5 and x² < 400}.

List all the elements of each of the sets given below.

A = {x : x = 2n, n ϵ N and n ≤ 5}.

List all the elements of each of the sets given below.

B = {x : x = 2n + 1, n ϵ W and n ≤ 5}.

List all the elements of each of the sets given below.

List all the elements of each of the sets given below.

D = {x : x = n², n ϵ N and 2 ≤ n ≤ 5}.

List all the elements of each of the sets given below.

E = {x : x ϵ Z and x² = x}.

List all the elements of each of the sets given below.

.

List all the elements of each of the sets given below.

List all the elements of each of the sets given below.

H = {x : x ϵ Z, |x| ≤ 2}.

Write each of the sets given below in set7builder from:

(i) .

(ii) .

(iii) C = {53, 59, 61, 67, 71, 73, 79}.

(iv) D = {–1, 1}.

(v) E = {14, 21, 28, 35, 42, …., 98}.

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

Which of the following are examples of the null set?

Set of odd natural numbers divisible by 2.

Which of the following are examples of the null set?

Set of even prime numbers.

Which of the following are examples of the null set?

A = {x : x ϵ N, 1 < x ≤ 2}.

Which of the following are examples of the null set?

B = {x : x ϵ N, 2x + 3 = 4}.

Which of the following are examples of the null set?

C = {x : x is prime, 90 < x < 96}.

Which of the following are examples of the null set?

D = {x : x ϵ N, x² + 1 = 0}.

Which of the following are examples of the null set?

E = {x : x ϵ W, x + 3 ≤ 3}.

Which of the following are examples of the null set?

F = {x : x ϵ Q, 1 < x < 2}.

Which of the following are examples of the null set?

G = {0}

Which of the following are examples of the singleton set?

(i) {x : x ϵ Z, x² = 4}.

(ii) {x : x ϵ Z, x + 5 = 0}.

(iii) {x : x ϵ Z, |x| = 1}.

(iv) {x : x ϵ N, x² = 16}.

(v) {x : x is an even prime number}

Which of the following are pairs of equal sets?

A = set of letters in the word, ‘ALLOY.’

B = set of letters in the word, ‘LOYAL.’

Which of the following are pairs of equal sets?

C = set of letters in the word, ‘CATARACT.’

D = set of letters in the word, ‘TRACT.’

Which of the following are pairs of equal sets?

E = {x : x ϵ Z, x² ≤ 4} and F = {x : x ϵ Z, x² = 4}.

Which of the following are pairs of equal sets?

G = {–1, 1} and H = {x : x ϵ Z, x² – 1 = 0}.

Which of the following are pairs of equal sets?

J = {2, 3} and K = {x : x ϵ Z, (x² + 5x + 6) = 0}

Which of the following are pairs of equivalent sets?

(i) A = {–1, –2, 0} and B = { 1, 2, 3,}

(ii) C = {x : x ϵ N, x < 3} and D ={x : x ϵ W, x < 3}

(iii) E = {a, e, i, o, u} and F = {p, q, r, s, t}

State whether any given set is finite or infinite:

A = Set of all triangles in a plane.

State whether any given set is finite or infinite:

B = Set of all points on the circumference of a circle.

State whether any given set is finite or infinite:

C = set of all lines parallel to the y-axis

State whether any given set is finite or infinite:

D = set of all leaves on a tree

State whether any given set is finite or infinite:

E = set of all positive integers greater than 500

State whether any given set is finite or infinite:

F = {x ϵ R: 0 < x < 1].

State whether any given set is finite or infinite:

G = {x ϵ Z: x < 1].

State whether any given set is finite or infinite:

H = {x ϵ Z: –15 < x < 15].

State whether any given set is finite or infinite:

J = {x : x ϵ N and x is prime}.

State whether any given set is finite or infinite:

K = {x : x ϵ N and x is prime}.

State whether any given set is finite or infinite:

L = set of all circles passing through the origin (0, 0)

Rewrite the following statements using the set notation:

(i) a is an element of set A.

(ii) b is not an element of A.

(iii) A is an empty set and B is a nonempty set.

(iv) A number of elements in A is 6.

(v) 0 is a whole number but not a natural number.

State in each case whether A ⊂ B or A ⊄ B.

A = {0, 1, 2, 3,}, B = {1, 2, 3, 4, 5}

State in each case whether A ⊂ B or A ⊄ B.

A =ϕ, B = {0}

State in each case whether A ⊂ B or A ⊄ B.

A = {1, 2, 3,}, B = {1, 2, 4}

State in each case whether A ⊂ B or A ⊄ B.

A = {x : xϵZ, x² = 1}, B = {x : x ϵ N, x² = 1}

State in each case whether A ⊂ B or A ⊄ B.

A = {x : x is an even natural number,}, B = {x : x is an integer}

State in each case whether A ⊂ B or A ⊄ B.

A = {x : x is an integer}, B = {x: x is a rational number}

State in each case whether A ⊂ B or A ⊄ B.

A = {x : x is a real number,}, B = {x : x is a complex number}

State in each case whether A ⊂ B or A ⊄ B.

A = {x : x is an isosceles triangle in the plane}, B = {x : x is an equilateral triangle in the same plane}

State in each case whether A ⊂ B or A ⊄ B.

A = {x : x is a square in a plane}, B = {x : x is a rectangle in the same plane}

State in each case whether A ⊂ B or A ⊄ B.

A = {x : x is a triangle in a plane}, B = {x : x is a rectangle in the same plane}

State in each case whether A ⊂ B or A ⊄ B.

A = {x : x is an even natural number less than 8}, B = {x : x is a natural number which divides 32}

Examine whether the following statements are true of false:

(i) {a, b} ⊄{b, c, a}

(ii) {a}ϵ {a, b, c}

(iii) ϕ ⊂{a, b, c}

(iv) {a, e} ⊂{x : x is a vowel in the English alphabet}

(v) {x : x ϵ W, x + 5 = 5} =ϕ

(vi) a ϵ {{a}, b}

(vii) {a} ⊂ {{a}, b}

(viii) {b, c} ⊂{a, {b, c}}

(ix) {a, a, b, b} = {a, b}

(x) {a, b, a, b, a, b, ….} is an infinite set.

(xi) If A = set of all circles of unit radius in a plane and B = set of all circles in the same plane then A⊂B.

If A = {1} and B = {{1}, 2} then show that A ⊄ B.

Hint 1 ϵ A but 1⊄ B.

Write down all subsets of each of the following sets:

(i) A = {A}

(ii) B = {a, b}

(iii) C = {–2, 3}

(iv) D = {–1, 0, 1}

(v) E =ϕ

(vi) F = {2, {3}}

(vii) G = {3, 4, {5, 6}}

Express each of the following sets as an interval:

(i) A = {x : x ϵR, –4 < x < 0}

(ii) B = {x : x ϵ R, 0 ≤ x < 3}

(iii) C = {x : x ϵ R, 2 < x ≤ 6}

(iv) D = {x : x ϵ R, –5 ≤ x ≤ 2}

(v) E = {x : x ϵ R, –3 ≤ x < 2}

(vi) F = {x : x ϵ R, –2 ≤ x < 0}

Write each of the following intervals in the set-builder from:

(i) A = (–2, 3)

(ii) B = [4, 10]

(iii) C = [–1, 8)

(iv) D = (4, 9]

(v) E = [–10, 0)

(vi) F = (0, 5]

if A = {3, {4, 5}, 6} find which of the following statements are true.

(i) {4, 5} ⊄A

(ii) {4, 5} ϵA

(iii) {{4, 5}} ⊆A

(iv) 4ϵA

(v) {3} ⊆A

(vi) {ϕ} ⊆A

(vii) ϕ⊆A

(viii) {3, 4, 5} ⊆A

(ix) {3, 6} ⊆A

If A = {a, b, c}, find P(A) and n{P(A)}.

If A = {1, {2, 3}}, find P(A) and n {P(A)}.

If A = ϕ then find n{P(A)}.

If A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 8} then find the universal set.

Prove that A ⊆B, B⊆ C and C⊆ A ⇒A = C.

For any set A, prove that A⊆ϕ ⇔ A =ϕ

State whether the given statement is true false:

(i) If A ⊂ B and x∉B than x ∉A.

(ii) If A ⊆ϕ then A = ϕ

(iii) If A, B and C are three sets such than A ϵ B and B ⊂ C then A ⊂ C.

(iv) If A, B and C are three sets such than A ⊂B and B ϵ C then A ϵC.

(v) If A, B and C are three sets such that A ⊄B and B ⊄C then A ⊄C.

(vi) If A and B are sets such that x A and A ϵ B then x ϵ B.

If A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n}, find:

(i) A ∪ B

(ii) B ∪ C

(iii) B ∪ C

(iv) C ∩ A

(vi) A ∩ B

If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8} and C = {10, 11, 12, 13, 14}, find:

(i) A ∪ B

(ii) B ∪ C

(iii) A ∪ C

(iv) B ∪ D

(v) (A ∪ B) ∪ C

(vi) (A ∪B) ∩C

(vii) (A ∩ B) ∪D

(viii) (A ∩ B) ∪ (B ∩ C)

(ix) (A ∩ C) ∩ (C ∪ D

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

(i) A ∩ B

(ii) A ∩ C

(iii) B ∩ C

(iv) B ∩ D

(v) B ∩ (C ∪ D)

(vi) A ∩ (B ∪ C)

If A = {x : x ϵ N}, B = {x : x ϵ N and x is even), C = {x : x ϵ N and x is odd} and D = {x : x ϵ N and x is prime} then find:

(i) A ∩ B

(ii) A ∩ C

(iii) A ∩ D

(iv) B ∩ C

(v) B ∩ D

(vi) C ∩ D

If A = {2x : x ϵ N}, 1 ≤ x < 4}, B = {x + 2) : x ϵ N and 2 ≤ x < 5} and C = {x : x ϵ N and 4 < x < 8}, find:

(i) A ∩ B

(ii) A ∪ B

(iii) (A ∪ B) ∩ C

If A = {2, 4, 6, 8, 10, 12}, B = {3, 4, 5, 6, 7, 8, 10}, find:

(i) (A – B)

(ii) (B – A)

(iii) (A – B) ∪ (B – A)

If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}, find:

(i) A ∩ (B – C)

(ii) A – (B ∪ C)

(iii) A – (B ∩ C)

If R is the set of all real numbers and Q is the set of all rational numbers then what is the set (R – Q)?

If A = {2, 3, 5, 7, 11} and B = ϕ, find:

(i) A ∪ B

(ii) A ∩ B

If A and B are two sets such that A ⊆ B then find:

(i) A ∪ B

(ii) A ∩ B

Which of the following sets are pairs of disjoint sets? Justify your answer.

(i) A = {3, 4, 5, 6} and B = {2, 5, 7, 9}

(ii) C = {1, 2, 3, 4, 5} and D = {6, 7, 9, 11}

(iii) E = {x : x ϵ N, x is even and x < 8}

F = {x : x = 3n, n ϵ N, and x < 4}

(vi) G = {x : x ϵ N, x is even} and H {x : x ϵ N, x is prime}

(v) J = {x : x ϵ N, x is even} and K = {x : x ϵ N, x is odd}

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find:

(i) A’

(ii) B’

(iii) C’

(iv) (B’)’

(v) (A ∪ B)’

(vi) (A ∩ C)’

(vii) ( B – C)’

if U = {a, b, c} and A = {a, c, d, e} then verify that:

(i) (A ∪ B)’ = (A’ ∩ B’)

(ii) (A ∩ B)’ = (A’ ∪ B’)

if U is the universal set and A U then fill in the blanks.

(i) A ∪ A’ = ….

(ii) A ∩ A’ = ….

(iii) ϕ ‘∩ A = ….

(iv) U’ ∩ A = ….

If A = {a, b, c, d, e}, B = {a, c, e, g}, verify that:

(i) A ∪ B = B ∪A

(ii) A ∪ C = C ∪ A

(iii) B ∪ C = C ∪ B

(iv) A ∩ B = B ∩ A

(v) B ∩ C = C ∩ B

(vi) A ∩ C = C ∩ A

(vii) (A ∪ B ∪ C = A ∪ (B ∪ C)

(viii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

If A = {a, b, c, d, e}, B = {a, c, e, g}, and C = {b, e, f, g} verify that:

(i) A ∩ (B – C) = (A ∩ B) – (A ∩ C)

(ii) A – (B ∩ C) = (A – B) ∪ (A – C)

If A = {x : x ϵ N, x ≤ 7}, B = {x : x is prime, x < 8} and C = {x : x ϵ N, x is odd and x < 10}, verify that:

(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, and = {2, 3, 5, 7} verify that:

(i) (A ∪ B)’ = (A’ ∩ B’)

(ii) (A ∩ C)’ = (A’ ∪ B’)

Let A = {a, b, c}, B = {b, c, d, e} and = {c, d, e, f} be subsets of U = {a, b, c, d, e, f}. Then verify that:

(i) (A’)’ = A

(ii) (A ∪ B)’ = (A’ ∩ B’)

(iii) (A ∩ B)’ = (A’ ∪ B’)

Given an example of three sets A, B, C such that A ∩ C ≠ ϕ, B ∩ C ≠ ϕ, A ∩ C ≠ ϕ, and A ∩ B ∩ C = ϕ

For any sets A and B, prove that:

(i) (A – B) ∩ B = ϕ

(ii) A ∪ (B – A) = A ∪ B

(iii) (A – B) ∪ (A ∩ B) = A

(iv) (A ∪ B) – B = A – B

(iv) A – (A ∪ B) = A – B

For any sets A and B, prove that:

(i) A ∩ B’ = ϕ ⇒ A B

(ii) A’ ∪ B’ = U ⇒ A B

Let A = {a, b, c, e, f} B = {c, d, e, g} and C = {b, c, f, g} be subsets of the set U = {a, b, c, d, e, f, g, h}.

(i) A ∩ B

(ii) A ∪ (B ∩ C)

(iii) A – B

(iv) B – A

(v) A – (B ∩ C)

(vi) (B – C) ∪ (C – B)

Let A = {2, 4, 6, 8, 10}, B = {4, 8, 12, 16} and C = {6, 12, 18, 24}.

Using Venn diagrams, verify that:

(i) (A ∪ B) ∪ C = A ∪ (B ∪ C)

(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C).

Let A = {a, e, I, o, u}, B = {a, d, e, o, v} and C = {e, o, t, m}.

Using Venn diagrams, verify the following

(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Let A _⊂ B ⊂ U. Exhibit it in a Venn diagram.

Let A = {2, 3, 5, 7, 11, 13}, B = {5, 7, 9, 11, 15} be subsets of U = {2, 3, 5, 7, 9, 11, 13, 15}.

Using Venn diagrams, verify that:

(i) (A ∪ B’) = (A’ ∩ B’)

(ii) (A ∩ B)’ = (A’ ∪ B’)

Using Venn diagrams, show that (A – B), A ∩ B) and (B – A) are disjoint sets, taking A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12, 15, }.

If A and B are two sets such that n(A) = 37, n(B) = 26 and n (A ∪ B) = 51, find n (A ∩ B).

If P and Q are two sets such that n(P ∪ Q) = 75, n (P ∩ Q) = 17 and n(P) = 49, find n (Q).

If A and B are two sets such that n(A) = 24, n(B) = 22 and n(A ∩ B) = 8, find:

(i) n(A ∪ B)

(ii) n(A – B)

(iii) n(B – A)

If A and B are two sets such that n(A –B) = 24, n(B – A) = 19 and n(A ∩ B) = 11, find:

(i) n(A)

(ii) n(B)

(iii) n(A ∪ B)

In a committee, 50 people speak Hindi, 20 speak English and 10 speak both Hindi and English. How many speak at least one of these two languages?

In a group of 50 persons, 30 like tea, 25 like coffee and 16 like both. How many like

(i) either tea or coffee?

(ii) neither tea nor coffee?

There are 200 individuals with a skin disorder, 120 had been exposed to the chemical C₁, 50 to chemical C₂, and 30 to both the chemicals C₁ and C₂. Find the number of individuals exposed to

(i) Chemical C₁ but not chemical C₂

(ii) Chemical C₂ but not chemical C₁

(iii) Chemical C₁ or chemical C₂

In a class of a certain school, 50 students, offered mathematics, 42 offered biology and 24 offered both the subjects. Find the number of students offering

(i) mathematics only,

(ii) biology only,

(iii) any of the two subjects.

In an examination, 56% of the candidates failed in English and 48% failed in science. If 18% failed in both English and science, find the percentage of those who passed in both the subjects.

In a group of 65 people, 40 like cricket and 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

A school awarded 42 medals in hockey, 18 in basketball and 23 in cricket. if these medals were bagged by a total of 65 students and only 4 students got medals in all the three sports, how many students received medals in exactly two of the three sports?

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read the newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, and 3 read all the three newspapers. Find

(i) The number of people who read at least one of the newspapers,

(ii) The number of people who read exactly one newspaper.

In a survey of 100 students, the number of students studying the various languages is found as English only 18; English but not Hindi 23; English and Sanskrit 8; Sanskrit and Hindi 8; English 26; Sanskrit 48 and no language 24. Find

(i) how many students are studying Hindi?

(ii) how many students are studying English and Hindi both?

In a town of 10,000 families, it was found that 40% of the families buy newspaper A, 20% buy newspaper B, 10% buy newspaper C, 5% buy A and B; 3% buy B and C, and 4% buy A and C. IF 2% buy all the three newspapers, find the number of families which buy

(i) A only,

(ii) B only,

(iii) none of A, B, and C.

A class has 175 students. The following description gives the number of students one or more of the subjects in this class: mathematics 100, physics 70, chemistry 46, mathematics and physics 30; mathematics and chemistry 28; physics and chemistry 23; mathematics, physics and chemistry 18. Find

(i) how many students are enrolled in mathematics alone, physics alone and chemistry alone,

(ii) The number of students who have not offered any of these subjects.

If a set A and n elements then find the number of elements in its power set P(A).

If A = ϕ then write P(A).

If n(A) = 3 and n(B) = 5, find:

(i) The maximum number of elements in A ∪ B,

(ii) The minimum number of elements in A ∪ B.

If A and B are two sets such than n(A) = 8, n(B) = 11 and n(A ∪ B) = 14 then find n(A ∩ B).

If A and B are two sets such that n(A) = 23, n(b) = 37 and n(A – B) = 8 then find n(A ∪ B).

If A and B are two sets such than n(A) = 54, n(B) = 39 and n(B – A) = 13 then find n(A ∪ B).

Hint n(B) = n(B – A) + n(A ∩ B) ⇒ n(A ∩ B) = (39 – 13) = 26.

If A ⊂ B, prove that B’ ⊂ A’.

If A ⊂ B, show that (B’ – A’) = ϕ.

Let A = {x : x = 6n N) and B = {x : x = 9n, n ϵ N}, find A ∩ B.

If A = {5, 6, 7}, find P(A).

If A = {3, {2}}, find P(A).

Prove that A ∩ (A B)’ = ϕ

Find the symmetric difference A Δ B, when A = {1, 2, 3} and B = {3, 4, 5}.

Prove that A – B = A ∩ B.’

If A = {x : x ϵ R, x < 5} and B = {x : x ϵ R, x > 4}, find A ∩ B.