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What is the difference between a collection and a set? Give reasons to support your answer.
Which of the following collections are sets? Justify your answer:
(i) A collection of all natural numbers less than 50.
(ii) The collection of good hockey players in India.
(iii) The collection of all the girls in your class.
(iv) The collection of all talented writers of India.
(v) The collection of difficult topics in Mathematics.
(vi) The collection of novels written by Munshi Prem Chand.
(vii) The collection of all questions of this chapter.
(viii) The collection of all months of a year beginning with the letter J.
(ix) A collection of most dangerous animals of the world.
(x) The collection of prime integers.
If A={0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol or in each of the following blank spaces:
i. 4 ……A ii. -4……A
iii. 12…..A iv. 9…..A
v. 0……A vi. -2……A
Describe the following sets in Roster form:
{x:x is a letter before e in the English alphabet}.
{x N: x2< 25}.
{x N: x is a prime number, 10 < x < 20}
{x N: x = 2n, n N}.
{x R: x > x}.
{x:x is a prime number which is a divisor of 60}.
{x:x is a two digit number such that the sum of its digits is 8}
The set of all letters in the word ‘Trigonometry’
The set of all letters in the word ‘Better.’
Describe the following sets in set-builder form:
(i) A = {1,2,3,4,5,6}
(ii) B={1, 1/2 , 1/3, 1/4, 1/5,…..}
(iii) C={0,3,6,9,12,….}
(iv) D={10,11,12,13,14,15}
(v) E={0}
(vi) {1,4,9,16,…,100}
(vii) {2,4,6,8,….}
(viii) {5,25,125,625}
List all the elements of the following sets:
A={x:x2≤ 10, xZ}
C=
D={x:x is a vowel in the word “EQUATION”}
e={x:x is a month of a year not having 31 days}
F={x:x is a letter of the word “MISSISSIPPI”}
Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form:
i.{A,P,L,E} i. {x:x+5=5, x z}
ii.{5,-5} ii. {x:x is a prime natural number and a divisor of 10}
iii. {0} iii. {x:x is a letter of the word “RAJASTHAN”}
iv. {1,2,5,10} iv. {x:x is a natural and divisor of 10}
v. {A,H,j,R,S,T,N} v. {x:x2 – 25 =0}
vi.{2,5} vi. {x:x is a letter of word “APPLE”}
Write the set of all vowels in the English alphabet which precede q.
Write the set of all positive integers whose cube is odd.
Write the set in the set-builder form.
Which of the following are examples of empty set?
(i) Set of all even natural numbers divisible by 5.
(ii) Set of all even prime numbers.
(iii) {x:x2–2=0 and x is rational}.
(iv) {x:x is a natural number, x < 8 and simultaneously x > 12}.
(v) {x:x is a point common to any two parallel lines}.
Which of the following sets are finite and which are infinite?
(i) Set of concentric circles in a plane.
(ii) Set of letters of the English Alphabets.
(iii) {x ∈ N : x > 5}
(iv) {x ∈ N : x < 200}
(v) {x ∈ Z: x < 5}
(vi) {x ∈ R:0 < x < 1}.
Which of the following sets are equal?
i. A = {1, 2, 3}
ii. B={x ∈ R:x2–2x+1=0}
iii. C= (1, 2, 2, 3}
iv. D={x ∈R : x3 – 6x2+11x – 6 = 0}.
Are the following sets equal?
A={x:x is a letter in the word reap}, B={x:x is a letter in the word paper},
C={x:x is a letter in the word rope}.
From the sets given below, pair the equivalent sets:
A= {1, 2, 3}, B={t, p, q, r, s}, C={α, β, γ}, D={a, e, I, o, u}.
Are the following pairs of sets equal? Give reasons.
A={2, 3}, B = {x : x is a solution of x2 + 5x + 6= 0}
A={x:x is a letter of the word “WOLF”}
B={x:x is letter of word “FOLLOW”}
From the sets given below, select equal sets and equivalent sets.
A={0, a}, B= {1, 2, 3, 4} C= {4, 8, 12},
D = {3, 1, 2, 4}, E={1, 0}, F = {8, 4, 12}
G={1, 5, 7, 11}, H={a, b}
A= {x:x ∈ N, x < 3},
B= {1, 2}, C= {3, 1}
D = {x : x ∈ N, x is odd, x < 5},
E= (1, 2, 1, 1}, F= {1, 1, 3}.
Show that the set of letters needed to spell “CATARACT” and the set of letters needed to spell “TRACT” are equal.
Which of the following statements are true? Give a reason to support your answer.
(i) For any two sets A and B either A B or B A.
(ii) Every subset of an infinite set is infinite.
(iii) Every subset of a finite set is finite.
(iv) Every set has a proper subset.
(v) {a,b,a,b,a,b,….} is an infinite set.
(vi) {a,b,c} and {1,2,3} are equivalent sets.
(vii) A set can have infinitely many subsets.
State whether the following statements are true or false:
(i) 1 { 1,2,3}
(ii) a ⊂ {b,c,a}
(iii) {a} {a,b,c}
(iv) {a,b} = {a,a,b,b,a}
(v) The set {x:x + 8 = 8} is the null set.
Decide among the following sets, which are subsets of which:
A = {x:x satisfies x2 – 8x + 12=0}
B= {2,4,6}
C= {2,4,6,8,….}
D={6}
Write which of the following statements are true? Justify your answer.
(i) The set of all integers is contained in the set of all rational numbers.
(ii) The set of all crows is contained in the set of all birds.
(iii) The set of all rectangles is contained in the set of all squares.
(iv) The set of all rectangle is contained in the set of all squares.
(v) The sets P={a} and B = {{a}} are equal.
(vi) The sets A={x:x is a letter of word “LITTLE”} AND, b = {x:x is a letter of the word “TITLE” } are equal.
Which of the following statements are correct? Write a correct form of each of the incorrect statements.
(i) a ⊂ {a,b,c}
(ii) {a} {a,b,c}
(iii) a {{a},b}
(iv) {a} ⊂ {{a},b}
(v) {b,c} ⊂ {a,{b,c}}
(vi) {a,b} ⊂ {a,{b,c}}
(vii) ϕ {a,b}
(viii) ϕ ⊂ {a,b,c}
(ix) {x:x + 3 = 3}= ϕ
Let A={a,b,{c,d},e}. Which of the following statements are false and why?
(i) {c,d} ⊂ A
(ii) {c, d} A
(iii) {{c,d}} ⊂ A
(iv) a A
(v) a ⊂ A.
(vi) {a,b,e} ⊂ A
(vii) {a,b,e} A
(viii) {a,b,c} ⊂ A
(ix) ϕ A
(x) {ϕ} ⊂ A
Let A = {{1,2,3}, {4,5}, {6,7,8}}. Determine which of the following is true or false:
(i) 1 A
(ii) {1,2,3} ⊂ A
(iii) {6,7,8} A
(iv) {2, ϕ} ⊂ A
(v) 2 ⊂ A
(vi) {2,{1}} A
(vii) {{2}}, {1}} A
(viii) {ϕ,{ϕ}, {1, ϕ}} ⊂ A.
Let A = {ϕ, {ϕ}, 1, {1, ϕ}, 2}. Which of the following are true?
(i) ϕ ∈ A
(ii) {ϕ} ∈ A
(iii) {1} ∈ A
(vi) {2, {1}} ⊄A
(vii) {{2}, {1}} ⊄ A
(viii) {ϕ, {ϕ}, {1, ϕ}} ⊂ A
(ix) {{ϕ}} ⊂ A.
Write down all possible subsets of each of the following sets:
Total number of the subset of any given set =2n
(i) {a}
(ii) {0,1}
(iii) {a,b,c}
(iv) {1,{1}}
(v) {ϕ}
Write down all possible proper subsets each of the following sets.
(i) {1,2}
(ii) {1,2,3}
(iii) {1}
What is the total number of proper subsets of a set consisting of n elements?
If A is any set, prove that: A ϕ A=ϕ.
Prove that: A B, B C and C A A = C.
How many elements have P (A), if A= ϕ?
What universal set (s) would you propose for each of the following:
(i) The set of right triangles.
(ii) The set of isosceles triangles
If A and B are two sets such that A ⊂ B, then Find:
A ∩ B
A ∪ B
If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13 , 14}. Find:
i. A ∪ B
ii. A ∪ C
iii. B ∪ C
iv. B ∪ D
v. A ∪ B ∪ C
vi. A ∪ B ∪ D
vii. B ∪ C ∪ D
viii. A ∩ (B ∪ C)
ix. (A ∩ B) ∩ (B ∩ C)
x. (A ∪ D) ∩ (B ∪ C).
Let A = {x:x ∈ N}, B = {x:x = 2n, n ∈ N), C = {x:x = 2n – 1, n ∈ N} and, D = {x:x is a prime natural number} Find:
i. A ∩ B
ii. A ∩ C
iii. A ∩ D
iv. B ∩ C
v. B ∩ D
vi. C ∩ D
Let A = {3, 6, 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}.
Find:
i. A – B
ii. A – C
iii. A – D
iv. B – A
v. C – A
vi. D – A
vii. B – C
viii. B – D
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:
A’
B’
(A ∩ C)’
(A∪B)’
(A’)’
(B – C)’
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that:
(A ∪ B)’ = A’ ∩ B’
(A ∩ B}’ = A’ ∪ B’.
Find the smallest set A such that A ∪ {1, 2} = {1, 2, 3, 5, 9}.
Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities:
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∩ (B – C) = (A ∩ B) – (A ∩ C)
A– (B ∪ C) = (A – B) ∩ (A–C)
A– (B ∩ C) = (A – B) ∪ (A – C)
A ∩ (B ∪ C) = (A ∩ B) ∩ (A ∩ C).
If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:
For any two sets A and B, prove that
B ⊂ A ∪ B
A ∩ B ⊂ A
A ⊂ B A ∩ B = A
For any two sets A and B, show that the following statements are equivalent:
A ⊂ B
A – B = ϕ
A ∪ B = B
A ∩ B = A.
For three sets A, B, and C, show that
A ∩ B = A ∩ C need not imply B = C.
A ⊂ B C – B ⊂ C – A
For any two sets, prove that:
A ∪ (A ∩ B) = A
A ∩ (A ∪ B) = A
Find sets A, B, and C such that A ∩ B, A ∩ C and B ∩ C are non–empty sets and A ∩ B ∩ C = ϕ.
For any two sets A and B, prove that: A ∩ B = ϕ A B’.
If A and B are sets, then prove that A – B, A ∩ B and B – A are pair wise disjoint.
Using properties of sets, show that for any two sets A and B, (A ∪ B) ∩ (A ∪ B’) = A.
For any two sets of A and B, prove that:
A’ ∪ B = U A ⊂ B
B’ ⊂ A’ 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)
A ∪ (B – A) = A ∪ B
For any two sets A and B, prove that: A’ – B’ = B – A
For any two sets A and B, prove the following:
A ∩ (A’ ∪ B) = A ∩ B
A – (A – B) = A ∩ B
A ∩ (A ∪ B’) = ϕ
A – B = A Δ (A ∩ B)
If A, B, C are three sets such that A ⊂ B, then prove that C – B ⊂ C – A.
(A ∪ B) – B = A – B
A – (A ∩ B) = A – B
(A – B) ∪ (A ∩ B) = A
If A and B are two sets such that n (A ∪ B) = 50, n(A) = 28 and n(B) = 32, find n (A ∩ B).
If P and Q are two sets such that P has 40 elements, P ∪ Q has 60 elements and P ∩ Q has 10 elements, how many elements does Q have?
In a school, there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics, and 4 teach physics and mathematics. How many teach physics?
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 like both coffee and tea?
Let A and B be two sets such that : n (A) = 20, n (A ∪ B) = 42 and n (A ∩ B) = 4. Find
n (B)
n (A – B)
n (B – A)
A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?
In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:
i. How many can speak both Hindi and English.
ii. How many can speak Hindi only.
iii. how many can speak English only.
In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find:
i. how may drink tea and coffee both.
ii. how many drink coffee but not tea.
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:
the numbers of people who read at least one of the newspapers.
the number of people who read exactly one newspaper.
Of the members of three athletic teams in a certain school, 21 are in the basketball team, 26 in the hockey team and 29 in the football team. 14 play hockey and basket ball, 15 play hockey and football, 12 play football and basketball and 8 play all the three games. How many members are there in all?
In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali?
A survey of 500 television viewers produced the following information;285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games? How many watch exactly one of the three games?
In a survey of 100 persons, it was found that 28 read magazine A, 30 read magazine B, 42 read magazine C, 8 read magazines A and B, 10 read magazines A and C, 5 read magazines B and C and 3 read all the three magazines. Find:
How many read none of three magazines?
How many read magazine C only?
In a survey of 100 students, the number of students studying the various languages was found to be: English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find:
How many students were studying Hindi?
How many students were studying English and Hindi?
In a survey it was found that 21 persons liked product P1, 26 liked product P2 and 29 liked product P3. If 14 persons liked products P1 and P2; 12 persons liked product P3 and P1; 14 persons liked products P2, and P3 and 8 liked all the three products. Find how many liked product P3 only.
If a set contains n elements, then write the number of elements in its power set.
Write the number of elements in the power set of null set.
Let A = {x : xN, x is a multiple of 3} and B = {x : xN and x is a multiple of 5}. Write .
Let A and B be two sets having 3 and 6 elements respectively. Write the minimum number of elements that can have.
If A = {xC : x2 = 1} and B = {xC : x4 = 1}, then write A − B and B − A.
If A and B are two sets such that , then write B’ − A’ in terms of A and B.
Let A and B be two sets having 4 and 7 elements respectively. Then write the maximum number of elements that can be.
If and , then write .
If A = {(x, y) : y = ex, xR} and B = {(x, y) : y = e−x, xR}, then write .
If A and B are two sets such that n(A) = 20, n(B) = 25 and n= 40, then write n.
If A and B are two sets such that n(A) = 115, n(B) = 326, n (A − B) = 47, then write .
Mark the correct alternative in the following:
For any set A, (A’) is equal to
Let A and B be two sets in the same universal set. Then, A − B =
The number of subsets of a set containing n elements is
For any two sets A and B ,=
If A = {1, 3, 5, B} and B = {2, 4}, then
The symmetric difference of A and B is not equal to
The symmetric difference of A = {1, 2, 3} and B = {3, 4, 5} is
For any two sets A and B, (A − B)(B − A) =
Which of the following statement is false:
For any three sets A, B and C
Let A = {x : xR, x > 4} and B = {xR : x < 5}. The, =
Let be the universal set containing 700 elements. If A, B are sub-sets of such that n (A) = 200, n (B) = 300 and n=100. Then, n=
Let A and B be two sets such that n (A) = 16, n (B) = 25. Then, n is equal to
If A = {1, 2, 3, 4, 5}, then the number of proper subsets of A is
In set-builder method the null set is represented by
If A and B are two disjoint sets, then n is equal to
For two sets iff
If A and B are two sets such that n(A) = 70, n(B) = 60, n= 110, then n is equal to
If A and B are two given sets, then is equal to
If A = {x : x is a multiple of 3} and, B = {x : x is a multiple 5}, then A − B is
In a city 20% of the population travels by car, 50% travels by bus and 10% travels by both car and bus. Then, persons travelling by car or bus is
If , then
An investigator interviewed 100 students to determine the performance of three drinks: milk, coffee and tea. The investigator reporter that 10 students take all three drinks milk, coffee and tea; 20 students take milk coffee; 25 students take milk and tea; 20 students take coffee and tea; 12 students take milk only; 5 students take coffee only and 8 students take tea only. Then the number of students who did not take any of three drinks is
Two finite sets have m and n elements. The number of elements in the power set of first set is 48 more than the total number of elements in power in power set of the second set. Then, the values of m and n are:
In a class of 175 students the following data shows the number of students opting 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?
Suppose A1, A2,...,A30 are thirty sets each having 5 elements and B1, B2,...,Bn are n sets each with 3 elements, let and each element of S belongs to exactly 10 of the and exactly 9 of the , then n is equal to
Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second. The values of m and n are respectively.
For any two sets A and B, is equal to
The set is equal to
Let F4 be the set of all parallelograms, F2 the set of all rectangles, F3 the set of all rhombuses, F4 the set of all squares and F5 the set of trapeziums in a plane. Then F1 may be equal to
