Which of the following are sets? Justify your answer.

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

(2) The collection of ten most talented writers of India.

(3) A team of eleven best-cricket batsmen of the world.

(4) The collection of all boys in your class.

(5) The collection of all natural numbers less than 100.

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

(7) The collection of all even integers.

(8) The collection of questions in this Chapter.

(9) A collection of most dangerous animals of the world.

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

(1) 5...A           

(2) 8...A             

(3) 0...A           

(4) 4...A          

(5) 2...A                

(6) 10...A

Write the following sets in roster form:

(1) A = {x: x is an integer and –3 < x < 7}.

(2) B = {x: x is a natural number less than 6}.

(3) C = {x: x is a two-digit natural number such that the sum of its digits is 8}

(4) D = {x: x is a prime number which is divisor of 60}.

(5) E = The set of all letters in the word TRIGONOMETRY.

(6) F = The set of all letters in the word BETTER.

Write the following sets in the set-builder form:

(1) (3, 6, 9, 12)                       

(2) {2, 4, 8, 16, 32}                      

(3) {5, 25, 125, 625}

(4) {2, 4, 6 ...}                       

(5) {1, 4, 9 ... 100}

List all the elements of the following sets:

(1) A = {x: x is an odd natural number}

(2) B = {x: x is an integer, -  < x < }

(3) C = {x: x is an integer, x² ≤ 4}

(4) D = {x: x is a letter in the word “LOYAL”}

(5) E = {x: x is a month of a year not having 31 days}

(6) F = {x: x is a consonant in the English alphabet which proceeds k}.

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

(1) {1, 2, 3, 4} - (a) {x: x is a prime number and divisor of 6}

(2) {2, 3} - (b) {x: x is an odd number less than 10}

(3) {M, A, T, H, E, I, C, S} - (c) {x: x is natural number and divisor of 6}

(4) {1, 3, 5, 7, 9} - (d) {x: x is a letter of the word MATHEMATICS}

Which of the following are examples of the null set

(1) Set of odd natural numbers divisible by 2

(2) Set of even prime numbers

(3) {x: x is a natural number, x < 5 and x > 7}

(4) {y: y is a point common to any two parallel lines}

Which of the following sets are finite or infinite

(1) The set of months of a year

(2) {1, 2, 3, . . .}

(3) {1, 2, 3, . . .99, 100}

(4) The set of positive integers greater than 100

(5) The set of prime numbers less than 99

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

(1) The set of lines which are parallel to the x-axis

(2) The set of letters in the English alphabet

(3) The set of numbers which are multiple of 5

(4) The set of animals living on the earth

(5) The set of circles passing through the origin (0, 0)

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

(1) A = {a, b, c, d}; B = {d, c, b, a}

(2) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}

(3) A = {2, 4, 6, 8, 10}; B = {x: x is positive even integer and x ≤ 10}

(4) A = {x: x is a multiple of 10}; B = {10, 15, 20, 25, 30, . . .}

Are the following pair of sets equal? Give reasons.

(1) A = {2, 3}, B = {x: x is solution of x² + 5x + 6 = 0}

(2) A = {x: x is a letter in the word FOLLOW}

B = {y: y is a letter in the word WOLF}

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}

Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:

(1) {2, 3, 4} . . . {1, 2, 3, 4,5}

(2) {a, b, c} . . . {b, c, d}

(3) {x: x is a student of Class XI of your school} . . . {x: x student of your school}

(4) {x: x is a circle in the plane} . . . {x: x is a circle in the same plane with radius 1 unit}

(5) {x: x is a triangle in a plane} . . . {x: x is a rectangle in the plane}

(6) {x: x is an equilateral triangle in a plane} . . . {x: x is a triangle in the same plane}

(7) {x: x is an even natural number} . . . {x: x is an integer}

Examine whether the following statements are true or false:

(1) {a, b} ⊄ {b, c, a}

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

(3) {1, 2, 3} ⊂ {1, 3, 5}

(4) {a} ⊂ {a, b, c}

(5) {a} ∈ {a, b, c}

(6) {x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}

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

(1) {3, 4} ⊂ A                 

(2) {3, 4} ∈ A                

(3) {{3, 4}} ⊂ A         

(4) 1 ∈ A              

(5) 1 ⊂ A

(6) {1, 2, 5} ⊂ A          

(7) {1, 2, 5} ∈ A         

(8) {1, 2, 3} ⊂ A       

(9) φ ∈ A

(10) φ ⊂ A

(11) {φ} ⊂ A

Write down all the subsets of the following sets

(1) {a}                         

(2) {a, b}                       

(3) {1, 2, 3}                      

(4) φ

How many elements has P(A), if A = φ?

Write the following as intervals:

(1) {x: x ∈ R, – 4 < x ≤ 6}                           

(2) {x: x ∈ R, – 12 < x < –10}

(3) {x: x ∈ R, 0 ≤ x < 7}                            

(4) {x: x ∈ R, 3 ≤ x ≤ 4}

Write the following intervals in set-builder form:

(1) (–3, 0)                        

(2) [6, 12]                     

(3) (6, 12]                            

(4) [–23, 5)

What universal set(s) would you propose for each of the following:

(1) The set of right triangles.                       

(2) The set of isosceles triangles.

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

(1) {0, 1, 2, 3, 4, 5, 6}                                       

(2) φ

(3) {0,1,2,3,4,5,6,7,8,9,10}                           

(4) {1,2,3,4,5,6,7,8}

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

(1) X = {1, 3, 5}; Y = {1, 2, 3}

(2) A = [a, e, i, o, u}; B = {a, b, c}

(3) A = {x: x is a natural number and multiple of 3}

B = {x: x is a natural number less than 6}

(4) A = {x: x is a natural number and 1 < x ≤6}

B = {x: x is a natural number and 6 < x < 10}

(5) A = {1, 2, 3}, B = φ

Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?

If A and B are two sets such that A ⊂ B, then what is A ∪ B?

Prove that: {cos (π + x) cos(-x) ÷ {sin (π - x) cos (  + x)} = cot² x

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

(1) A ∪ B                  

(2) A ∪ C                 

(3) B ∪ C             

(4) B ∪ D              

(5) A ∪ B ∪ C

(6) A ∪ B ∪ D       

(7) B ∪ C ∪ D

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

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

(1) A ∩ B                  

(2) B ∩ C                 

(3) A ∩ C ∩ D               

(4) A ∩ C             

(5) B ∩ D

(6) A ∩ (B ∪ C)      

(7) A ∩ D              

(8) A ∩ (B ∪ D)           

(9) (A ∩ B) ∩ (B ∪ C)

(10) (A ∪ D) ∩ (B ∪ C)

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

C = {x: x is an odd natural number} and D = {x: x is a prime number}, find

(1) A ∩ B         

(2) A ∩ C          

(3) A ∩ D         

(4) B ∩ C            

(5) B ∩ D          

(6) C ∩ D

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

C = {x: x is an odd natural number} and D = {x: x is a prime number}, find

(1) A ∩ B         

(2) A ∩ C          

(3) A ∩ D         

(4) B ∩ C            

(5) B ∩ D          

(6) C ∩ D

Which of the following pairs of sets are disjoint

(1) {1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x ≤ 6}

(2) {a, e, i, o, u} and {c, d, e, f}

(3) {x: x is an even integer} and {x: x is an odd integer}

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

(1) A – B           

(2) A – C           

(3) A – D          

(4) B – A           

(5) C – A               

(6) D – A

(7) B – C       

(8) B – D         

(9) C – B           

(10) D – B           

(11) C – D              

(12) D – C

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

(1) X – Y                                  

(2) Y – X                                 

(3) X ∩ Y

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

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

(1) {2, 3, 4, 5} and {3, 6} are disjoint sets.

(2) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.

(3) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.

(4) {2, 6, 10} and {3, 7, 11} are disjoint sets.

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

(1) A′          

(2) B′       

(3) (A ∪ C) ′        

(4) (A ∪ B) ′           
(5) (A′) ′             

(6) (B – C) ′

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

(1) A = {a, b, c}              

(2) B = {d, e, f, g}            

(3) C = {a, c, e, g}             

(4) D = {f, g, h, a}

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

(1) {x: x is an even natural number}

(2) {x: x is an odd natural number}

(3) {x: x is a positive multiple of 3}               

(4) {x: x is a prime number}

(5) {x: x is a natural number divisible by 3 and 5}

(6) {x: x is a perfect square}                         

(7) {x: x is a perfect cube}

(8) {x: x + 5 = 8}                                            

(9) {x: 2x + 5 = 9}

(10) {x: x ≥ 7}                                                       

(11) {x: x ∈ N and 2x + 1 > 10}

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

(1) (A ∪ B) ′ = A′ ∩ B′                             

(2) (A ∩ B) ′ = A′ ∪ B′

Draw appropriate Venn diagram for each of the following:

(1) (A ∪ B) ′,          

(2) A′ ∩ B′,               

(3) (A ∩ B) ′,              

(4) A′ ∪ B′

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°, what is A′?

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

(1) A ∪ A′ = …….          

(2) φ′ ∩ A =……….       

(3) A ∩ A′ = ………       

(4) U′ ∩ A = ……….

If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩ Y).

If X and Y are two sets such that X ∪ Y has 18 elements, X has 8 elements and Y has 15 elements;

how many elements does X ∩ Y have?

In a group of 400 people, 250 can speak Hindi and 200 can speak English.

How many people can speak both Hindi and English?

If S and T are two sets such that S has 21 elements, T has 32 elements,

and S ∩ T has 11 elements, how many elements does S ∪ T have?

If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and X ∩ Y has 10 elements,

how many elements does Y have?

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks.

How many people like both coffee and tea?

In a group of 65 people, 40 like cricket, 10 like both cricket and tennis.

How many like tennis only and not cricket? How many like tennis?

In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French.

How many speak at least one of these two languages?

 

Decide, among the following sets, which sets are subsets of one and another:

A = {x: x ∈ R and x satisfy x² – 8x + 12 = 0}, 

B = {2, 4, 6},   

C = {2, 4, 6, 8, . . .},   

D = {6}.

In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(1) If x ∈ A and A ∈ B, then x ∈ B

(2) If A ⊂ B and B ∈ C, then A ∈ C

(3) If A ⊂ B and B ⊂ C, then A ⊂ C

(4) If A ⊄ B and B ⊄ C, then A ⊄ C

(5) If x ∈ A and A ⊄ B, then x ∈ B

(6) If A ⊂ B and x ∉ B, then x ∉ A

Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.

Show that the following four conditions are equivalent:

(1) A ⊂ B               

(2) A – B = φ                  

(3) A ∪ B = B                      

(4) A ∩ B = A

 

Show that if A ⊂ B, then C – B ⊂ C – A.

Assume that P(A) = P(B). Show that A = B

Is it true that for any sets A and B, P(A) ∪ P(B) = P (A ∪ B)? Justify your answer.

Show that for any sets A and B,

A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)

Using properties of sets, show that

(1) A ∪ (A ∩ B) = A                               

(2) A ∩ (A ∪ B) = A.

Show that A ∩ B = A ∩ C need not imply B = C.

Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X,

show that A = B. (Hints A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use Distributive law)

Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = φ.

In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee,

100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

In a group of students, 100 students know Hindi, 50 know English and 25 know both.

Each of the students knows either Hindi or English. How many students are there in the group?

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I,

9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:

(1) the number of people who read at least one of the newspapers.

(2) the number of people who read exactly one newspaper.

In a survey it was found that 21 people liked product A, 26 liked products B and 29 liked product C.

If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products.

Find how many liked product C only.