IMPORTANT SYMBOLS
Symbol Meaning
= equal to
≠ not equal to
< Less than
> Greater than
≤ less than or equal to
≥ greater than or equal to
< less than
> not greater than
∈ belongs to or is an element of
∉ does not belong to or not an element of
∨ or
∧ and
∀x for all values of x
: such that
| Such that
⇒ implies that
⊂ subset of or is contains in
⊂ Proper Subset of
⊄ not a subset of or is not contained in
) Superset of or contains
⊅ not a superset of or does not contain
- Difference
Aᶜ or A′ complement of set A
∪ union or cup
∩ intersection or cap
∅ null or void or empty set
U or S universal set
Z set of integers (0,±1,±2, ...)
Z⁺ set of positive integers (1,2,3..)
Z⁻ set of negative integers(-1,-2,..)
N set of natural numbers (1,2,3..)
Q set of rational number
Q⁺ set of positive rational numbers
Q⁻ set of negative rational numbers
R set of real numbers
R⁺ set of positive real numbers
R⁻ set of negative real numbers
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DEFINITION:
A well defined collection of objects is called a set.
When
1. the collection is well defined.
2. objects belonging to the collection are different.
3. objects of the collection are independent of the order of their arrangements
EXAMPLES OF SETS
Sn. Set Description
1. N The set of all natural numbers
2. Z The set of all integers.
3. Q The set of all rational nos.
4. R The set of all real nos.
5. Z⁺ The set of all positive integers.
6. Q⁺ The set of all positive rational numbers.
7. R⁺ The set of all positive real numbers.
• Sets are usually denoted by capital letters A,B,C,X,Y,Z etc.
• Members of a set: All the entries in a set are called it's it's it's members or elements or objects.
• The elements of a set are represented by small letters a,b,c,x,y,z etc.
• If a is an element of a set A, then we write ' a ∈ A '. We say it that ' a belongs to A'.
• If b is not an element of a set A, then we write 'b ∉ A'. We say it that 'b does not belong to A'
EXERCISE - A
Following collection are sets or not:
1) The collection of all vowels in the English alphabet. Y/N
2) The collection of all even natural numbers less than 10. Y/N
3) the collection of all prime number less than 20 Y/N
4) the collection of all beautiful girls of India. Y/N.
5) the collection of all the days of a week. Y/N
6) the collection of all the months of year beginning with the letter M. Y/N
7) a collection of 11 best Indian hockey players. Y/N
8) the collection of all rich persons of Delhi. Y/N
9) The collection of all persons of Chennai whose assessed annual income exceeds ₹ 10 lakhs. Y/N
10) the collection of all Books on mathematics. Y/N
11) Collection of all interesting books. Y/N.
12) the collection of all students of your class. Y/N
13) the collection of short boys of your class. Y/N.
14) the collection of all planets of the solar system. Y/N
15) The collection of five most talented writers of India. Y/N.
16) The collection of all interesting dramas written by Shakespeare. Y/N
17) the collection of all letters of the English alphabet which precedes K. Y/N
18) The collection of fat boys of your locality. Y/N.
19) the collection of all those students of your class whose age exceeds 15 years. Y/N
Answer:
1) yes 2) yes 3) yes 4) no 5) yes 6) yes 7) no 8) no 9) no 10) no 11) no 12) yes 13) no 14) yes 15) no 16) no 17) yes 18) no 19) yes
EXERCISE - B
Describe the following sets in Roster form:
1) (x:x is a letter before e in the English alphabet).
A) {a,b,c,d}. B){b,c,d} C){b,c} D) {b}
2) {x ∈N: x²< 25}.
A) {1,4,9,16} B){1,2,3,4}.
C) {1,4,9,16,25} D) {1,2,3,4,5}
3) {x ∈ N : is a prime number, 10<x< 20}.
A) {11,13,17,19}. B) {10,11,12,13}
C) {10, 13,17,19} D) {11,12,13,14}
4) { x∈N : x= 2n, n ∈N}.
A) {2,4,6,8,..}. B) {1,2,3,4}
C) {2,4,6,8} D) none
5) {x ∈R : x > x}.
A) φ. B) {0} C) {1} D) none
6) {x: x is a prime number which is a divisor of 60}.
A) {2,3,5}.
B) {1,2,3,4,5,6,10,12,15,30,60}
C) {10,12,15,30,60} D) none
7) {x : x is a two-digit number such that the sum of its digits is 8}.
A) {17,26,35,44,53,62,71,80}.
B) {1,2,3,4,5,6,6,7,8}
C) {1,2,3,4} D) none
8) the set of all letters in the word 'Trigonometry'.
A) {T,R,I,G,O,N,M,E,Y}.
B) {T,R,G,N,M,R}
C) {I,O,E} D) None
9) The set of all letters in the word 'Better'
A) {B,E,T,R} B) {B,e,t,r}
C) {B,E,T,E,R} D) none
10) {x: x+5=5, n∈Z}.
A) { } B) {0}. C) {φ} D) none
11) {x: x is a prime natural number and a divisor of 10}.
A) {1,2,5} B){2,5}. C) {1,2,5,10} D)n
12) {x: x is a natural number and divisor of 10}.
A){1,2,5,10} B){2,5} C){2,5,10} D)n
13) {x:x is a letter of the word "RAJASTHAN'}.
A) {A,H,J,R,S,T,N}.
B) {H,J,R,S,T,N}
C) {A,H,J,R,S,T,} D) n
14) {x:x² - 25= 0}.
A){-5,5,10} B) {5,-5}.
C) {1,5,-5,10} D) {0,-5,5}
15) {x:x is a letter of the word "APPLE"}.
A) {A,P,L,E}. B) {A,P,P,L,E} C) {A,E} D) none
Answer:
1) A 2.B 3. A 4.A 5.A 6.B 7.A 8.A 9.B 10.B 11.B 12.A 13A 14.B
EXERCISE - C
Describes the following shapes in set builder form:
1) P={1,2,3,4,5,6}.
A) {x:x ∈N,x<7}. B) {x:x ∈N,x≤7}
C) {x:x ∈I,x<7} D) none
2) Q={1, 1/2,1/3,1/4,1/5,...}.
A) {x:x= 1/n², x∈N}
B) {x:x= 1/n, x∈N}.
C) {x:x= n, x∈N} D) none
3) R={0,3,6,9,12}
A) {x: x=3n, n∈N }.
B) {x:x= n, x∈N}
C) {x:x= 1/n², x∈N}
D) {x:x= n², x∈N}
4) D={10,11,12,13,14,15}.
A) {x: x∈ N, 9≤ x<16}
B) {x: x∈ N, 9< x≤16}
C) {x: x∈ N, 9< x<16}.
D) {x: x∈ N, 9≤ x≤16}
5) E={0}.
A) {x: x=0}. B) {x: x<0}
C) {x: x>0} D) none
6) {1,4,9,16,...100}.
A) {x²: x ∈N, 1≤ x<10}
B) {x²: x ∈N, 1≤ x≤ 10}.
C) {x²: x ∈N, 1< x≤ 10}
D) {x²: x ∈N, 1< x< 10}
7) {2,4,6,8,...}.
A) {x: x=2n, n∈N}.
B) {x: x=n², n∈N}
C) {x: x=n²-1, n∈N}
D) {x: x=n³, n∈N}
8) {5,25,125,625}.
A) {5ⁿ: n∈N, 1≤n≤4}.
B) {5ⁿ: n∈N, 1<n≤4}
C) {5ⁿ: n∈N, 1≤n<4}
D) {5ⁿ: n∈N, 1<n<4}
EXERCISE - D
List all the elements of the following sets:
1) A={x: x²≤ 10, x∈ Z}.
A={0,±1,±2,±3}. B) {0,1,2,3}
C) {5,10} D) {1,5,10}
2) B={x: x= 1/(2n -1) , 1≤n≤5}.
A) B={1, 1/3,1/5,1/7,1/9}.
B) B={1/2, 1/3,1/5,1/7}
C) B={1/2, 1/3,1/4,1/5,1/6} D) n
3) C={x: x is an integer, -1/2<x<9/2}.
A) C={1,2,3,4}
B) C={0,1,2,3}
C) C={0,1,2,3,4}.
D) C={0,1,2,3,4,5}
4) D={ x:x is a vowel in the word 'EQUATION'}.
A) {A,E,I,O,U}. B) {Q,T,N}
C) {A,E,I,N,O,Q,T} D) none
5) E={x:x is a month of a year not having 31 days}.
A)E={Feb, April, June, Sept, November}.
B) {Jan, March, May, June, July, August, December}
C){Jan, March, June, August}
D) {February} E) none
6) F={x:x is a letter of the word 'MISSISSIPPI'}.
A) F={M,I,S,P}. B) F={M,S,P}
C) F=M,I,S,P D) F=(M,I,S,P)
7) Write the set of all vowels in the English alphabet which precedes q.
A) {a,e,i} B) {a,e,i,o}.C){a,e,i,o,u} D) n
8) Write the set of all positive integers whose cube is odd.
A) {2n+1: n∈Z, n ≥ 0}.
B) {1³,2³,3³,4³}
C) {1³,2³,3³,4³........} D) none
9) write the set {1/2, 2/5, 3/10, 4/17, 5/26, 6/37, 7/50} in the set builder form.
A) {n/(n²+1) : n∈N, n ≤7}.
B) {n/(n²+1) : n∈N, n <7}
C) {n/(n²+1) : n∈N, n >7}
D) {n/(n²+1) : n∈N, n >7}
Answer:
1.A 2.B 3.C 4.A 5.A 6.A 7B 8A 9A
EXERCISE - E
Rewrite the following statements, using set notations:
1) a is an element of set A.
A) a∈A. B) a= A C) a≠ A D) n(A)= a
2) b is not an element of A.
A) b ∉ A. B) a= A C) a≠ A D) n(A)= a
3) A is an empty set and B is a non empty set.
A) A= φ and B ≠ φ.
B) A≠ φ and B = φ
C) A≠ φ and B ≠ φ
D) A= φ and B = φ
4) number of element in A is 6.
A) n(A)= 6. B) A∈6 C) A ≠ 6 D) A=6
5) 0 is a whole number but not a natural number.
A) 0∈W but 0 ∉ N
B) 0∈N but 0 ∉ W
C) 0∈W but 0 ∉ W
D) 0∈N but 0 ∉ N
ANSWER:
1A 2A 3B 4A 5A
EXERCISE - F
State the type of set:
1) A= Set of all in triangles in a plane.
A) finite set B) Infinite set.
C) empty set D) Singleton set
2) B= set of all points on the circumference of a circle.
A) finite set B) Infinite set.
C) empty set D) Singleton set
3) C= set of all lines parallel to the y-axis.
A) finite set B) Infinite set.
C) empty set D) Singleton set
4) D= set of all leaves on a tree.
A) finite set. B) Infinite set
C) empty set D) Singleton set
5) E= set of all positive integers greater than 500.
A) finite set B) Infinite set.
C) empty set D) Singleton set
6) F={x ∈ R: 0<x < 1}
A) finite set B) Infinite set.
C) empty set D) Singleton set
7) G={x ∈ Z: x < 1}
A) finite set B) Infinite set.
C) empty set D) Singleton set
8) H={x ∈ Z : -15 < x < 15}
A) finite set. B) Infinite set
C) empty set D) Singleton set
9) I={x:x ∈ N and x² < 36}
A) finite set. B) Infinite set
C) empty set D) Singleton set
10) J={x: x ∈ Z and x < 10}
A) finite set B) Infinite set.
C) empty set D) Singleton set
11) K={x: x ∈ N and x is prime}.
A) finite set B) Infinite set.
C) empty set D) Singleton set
12) L={x:x ∈ N and x is even}.
A) finite set B) Infinite set.
C) empty set D) Singleton set
13) set of numbers which are multiple of 5.
A) finite set B) Infinite set.
C) empty set D) Singleton set
14) set of animals living on earth.
A) finite set. B) Infinite set
C) empty set D) Singleton set
15) M={x: x ∈ N and x >100}.
A) finite set B) Infinite set.
C) empty set D) Singleton set
16) N={x: x ∈ N and x < 1000}.
A) finite set. B) Infinite set
C) empty set D) Singleton set
17) A{x:x²-3= 0 and x is rational} is
A) finite B) infinite C) empty D) singleton set
18) B={x:x is an even prime number}.
A) finite B) infinite C) empty D) singleton set
19) C={x:4<x<5, x∈N}
A) finite B) infinite C) empty D) singleton set
20) M={x:x²= 25, and x is an odd integers}
A) finite B) infinite C) empty D) singleton set
21) set of all odd natural numbers divisible by 5.
A) finite B) infinite C) empty D) singleton set
22) set of all odd prime numbers.
A) finite B) infinite C) empty D) singleton set
23) set of all even prime numbers
A) finite B) infinite C) empty D) singleton set
24) S={x∈N, 2x+5= 6}
A) finite B) infinite C) empty D) singleton set
25) B={x|x ∈N, 1< x ≤2}
A) finite B) infinite C) empty D) singleton set
26) X={x|x is prime, 90< x ≤96}
A) finite B) infinite C) empty D) singleton set
27) B={x|x N, x² +4= 0}
A) finite B) infinite C) empty D) singleton set
28) K={0}
A) finite B) infinite C) empty D) singleton set
Answer:
1A 2B 3B 4B 5A 6C 7B 8A 9A 10A 11B 12B 13B 14B 15B 16B 17C 18D 19C 20A 21B 22B 23D 24C 25D 26C
EXERCISE - G
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}
A) i and ii B) ii and iii
C) iii and iv D) i and iii and iv.
2) which of the following sets are equal ?
i) A={x:x is a letter in the word reap}
ii) B={x:x is a letter in the word paper}
iii) C={x:x is a letter in the word rope}
A) ii and iii B) i and iii
C) i and ii D) i≠ ii≠ iii
3) A={1,2,3}; B={x,y,z} both set are
A) equal B) equivalent C) empty D) none
4) A={t,o,q,r,s}; B={a,e,i,o,u} both set are
A) equal B) equivalent C) empty D) none
5) A={2,3};
B={x:x is a solution of {x²+5x+6=0} both set are
A) equal B) equivalent C) empty D) none
6) A={x:x is a letter of the word 'WOLF'}
B={x:x is a letter of the word 'FOLLOW'}
A) equal B) equivalent C) empty D) none
7) A={1,2,3,4};
B={3,1,2,4} both set are
A) equal B) equivalent C) empty D) none
8) A={4,8,12};
B={8,4,12} both set are
A) equal B) equivalent C) empty D) none
9) A={0,a};
B={1,0} both set are
A) equal B) equivalent C) empty D) none
Answer:
1D 2C 3B 4B 5B 6A 7A 8A 9B
EXERCISE- H
List the members of the following sets :
1) The set of vowels.
2) A={x: x² ≤ 10, x ∈ Z}
3) The set of positive odd integers<10
4) B={x: x= 1/(2n-1) , 1 ≤n≤5}
5) The set of all integers lying between 6 and 12.
6) C={x: x is an integer, -1/2<x<9/2}
7) D={x: x is a vowel of MUMMY}
8) { x: x is a month of a year not having 31 days}
9) The set of all integers lying between 10 and 25 which are exactly divisible by 3.
10) The set of prime numbers<20.
11) The set of all positive integers whose cube is odd.
Answer:
1) {a, e, i, o,u} 2) {9,1,2,......10}
3) {1,3,5,7,9} 4) {1, 1/3, 1/5, 1/7, 1/9}
5) {7,8,9,10,11} 6) {0,1,2,3,4}
7) {M, U, Y}
8) {Feb, Apr, June, Sep, Nov}
9) {12,15,18,21,24}
10) {2,3,5,7,11,13} 11) {3,5,7}
REPRESENTATION OF A SET
1. Roster or Tabular Form:
All the members of the set are listed, the elements are being separated by commas and are enclosed within braces { }.
Example:
• The set of perfect square natural numbers less than or equal to 50 is represented by {1,4,9,25,36,49}
• W={0,1,2,3, ........}
EXERCISE - I
Write the sets in tabular form :
1) The set of all letters in the word of
i) MATTER
ii) MATHEMATICS
iii) EQUATION
2) The set of all natural numbers less than 7
3) The set of squares of integers
4) S= {x: is an integer and x<5}
5) {x:x is a letter before e in the English alphabet}
6) { x belongs to N: x² < 25}
7) S= {x:x is an integer and 5<x<25.
8) {x ∈N: x is prime number, 10<x<20}
9) {x∈R: x> x}
10) {x:x is a prime number which is a divisor of 60}
11) S={x: is an integer and -3≤x≤3}
12) {x:x is a two digit number such that the sum of its digits is 8}
13) S={x:x is an even integer}
14) S={x:x=2n+1 and n is an integer}
15) S= {x:x=4n and 10<x<25}
16) S={x:x² =9}
17) S ={x:x²+x-6=0}
18) S={x:x³-6x²+11x-6=0}
19) S={x:x is an even positive integer}
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2. Set builder Form:
One or more variables (say x,y,etc) are taken to represent all the properties possessed by every element of the set.
Example:
• The set of all real numbers greater than-2 and less than or equal to 1 can be written as {x∈R: -1<x ≤1)
• Q= x: x=m/n; where m,n are Integers and n≠0.
EXERCISE - J
Write down the following sets in symbolic/Set-Builder form :
1) The set of all letters in the word COMBINATION.
2) The set of reciprocals of natural numbers.
3)S={2,4,6,8….}
4) The set of all odd natural numbers.
5) The set of all even natural numbers.
6) A= {1,2, 3, 4, 5, 6}
7) S={5,10,15,20…}
8) B={1,1/2, 1/3,1/4, 1/5, ...}
9) S={1,4,7,10….}
10) C={0,3,6, 9, 12, ...}
11) D={10, 11, 12, 13, 14, 15}
12) P={0}
13) M={1,4,9,16,.....100}
14) {2, 4, 6, 8, .....}
15) {5, 25, 125, 625}
16) {1/2, 2/5, 3/10, 4/17, 5/26, 6/37, 7/50}
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TYPES OF SETE
Empty or Null or Void Set:
A set which contains no elements is called the empty set and is denoted by θ or { }
Example
• A= {x∈Z : 7 < x < 8}
• The set of all men each is 10 metres or more tall.
EXERCISE - K
State whether or not the following sets are empty:
1) The set of integers lying between 11 and 14 which are divisible by 5.
2) { x:x is an integer and 3< x<4}
3) The set of integers greater than 20 and less than 30 which are Divisible by 16.
4) A={x| x is even and x²= 25}
5) A={x: x is prime number and 32≤ x ≤36}
6) The set of prime numbers greater than 13 and less than 17.
7) Set of all natural numbers divisible by 5
8) S={x:x is a letter after z in the English alphabet.}
9) Set of all even prime numbers
10) S={x:x -1<x<1, x is an integer}
11) {x: x² -2=0 and x is rational}
12) S={x:x/2 +3=5, x is real}
13) {x: x is a point common to any two parallel lines }
14) S={x: x²=9, x is an even integer}
15) S={x: 4<x<5 and x is an integer}
16) S={x: x²+9=0, is real}
17) S={x: x²=4 and 2x-5}
18) S={x+4=4}
19) S={x: x≠x}
20) S={x:x²=2, x is rational}
Answer:
Empty set: 1,2,3,4,5,6,8,11,13,14,15,16,19,20
-----------:::::-------------:::::----------------
Singleton set:
A set containing only one element is called singleton set.
Example:
• The set of all natural satellites of the Earth.
• {4}
• X={x∈N, 1 < x <3}
Universal Set :
A set that contains all sets under consideration is called the universal set and is denoted by U.
Example:
If A= {1, 2, 3,4} and B={a,2,3,b}, then U= {1,2,3,4,a,b}
_____'''__________""___________
Finite Set:
A set is called a finite set if it is either an empty set or it contains a finite number of elements.
Example:
• The set of all negative integers more than-100.
• {a,b,c,d}
• A= {x:x is a vowel in the English alphabet}
Infinite Set:
A set is called an infinite set if it has infinite number of elements.
Example:
• The set of real numbers R
• {1,2,3,4,..}
• A= {x: x∈W}
• A set of all stars in the sky
EXERCISE- L
Are the following finite or infinite:
1) Set of all odd positive integers.
2) Set of concentric circles in a plane.
3) S={x:x is an integer}
4) Set of letters of the English alphabets.
5) {x ∈ N : x > 5}
6) S={x:x⁴-4x³+6x²-4x+1=0}
7) Set of all people living in India.
8) {x ∈N : x < 200}
9) Set of all rational numbers lying between 0 and 1.
10) S={x:x is an integer x≥4.
11) {x ∈ Z : x < 5}
12) Set of all lines in a plane passing through a fixed point (a,b)
13) { x ∈ R : 0 < x < 1}
14) The set of all people speaking Hindi.
___________________**__________________
• Cardinal Number of a Finite set:
The total number of members of a finite set A is called cardinal number of the set A and is denoted by n(A).
• Equal Sets :
Two or more sets having exactly the same elements are called equal sets. If the sets A and B are equal, then we write A= B.
For example, A= {m,n,h,t} and B={t,h,n,m} are equal sets. That is A= B.
• Equivalent Sets:
Two sets containing the same cardinal number are called equivalent sets.
For example, sets A={R,h,T,P} and B={4,12,90,102} are equivalent sets.
EXERCISE- M
Test whether the following pairs of sets are equal or not:
1) i) A={3,5,7,9}.
ii) B={the set of odd integers between 2 and 10}
2) A={1,2,3,4} & B={2,4,1,3}
3) A={1,4,9,16,25} &
B={the set of positive integers less than 30 which are perfect squares}.
4) i) A={1,2,3} &
ii)B={x: x³-6x+11x-6=9}
5) A={-1, -2} and D={x: x is a root of x²+ 3x+2=0}.
6) Find the equal sets from the following:
A={1,2,3};
B={x ∈R: x² - 2x+1=0};
C={1,2,2,3} ;
D={x ∈R: x³ - 6x²+11x-6=0}
7) 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}
8) A={x: x is a letter in REAP}
B={x:x is a letter PAPER}
C={x: x is a letter ROPE}
9) Find the equivalent sets:
A={1,2,3},
B={t,p,q,r,s},
C={x,y,z},
D={a,e,I,o,u}
10) A={2,3}, B={x:x is a solution of x²+5x+6=0}
11) A={x:x is a letter of WOLF}
B={ x:x is a letter of FOLLOW}
12) From the sets given, select equal 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}.
13) Show that the set of letters needed to spell CATARACT and the set of letters needed to spell TRACT are equal.
____________***______**_______
Subset:
A set X is said to be a subset of a set Y if every element of X is also an element of X is also an element of Y.
It is denoted by X⊂ Y
Example:
X={4,8,12} is a subset of Y={4,8,10,12,16}
Superset:
If X is a subset of a set Y, then Y is a super set of X.
Example:
If X={4,8,12}, then Y={4,8,12,16} is a superset of X.
Proper Subset:
If a set X contains some but not all the elements of a set Y, then X is a proper set of Y.
It is denoted by X⊂ Y.
Example:
If x={p,q,r} and Y={p,q,r,s,t}, then X⊂ Y.
***Proper Subset:
A superset X of a set Y that is not equal to Y, is called Proper Subset of Y.
Example:
If X={1,2,3,4,5,6} and Y={2,4,5}, then X is Proper Subset of Y.
Power Set:
A set whose all the elements are the subsets of a set A is called power set of A and is denoted by P(A)
Example:
If A={1,2}, then P(A)={θ, {1}, {2}, {1,2}}
EXERCISE - N
write all the subset:
1) A={1,3,5}
2) {2}
3) {0,1}
4) {1,{1}}
5) { }
_____________****_____________
OPERATION ON SETS
• Union of Sets :
The union of some sets is the set which consists of all the elements of these sets.
Union of two sets A and B is written as A ∪ B and read as 'A union B'
The symbol "∪" is used to denote "union".
• Intersection of Sets:
The intersection of some given sets is the set which contains only common elements to the given sets.
Intersection of two sets A and B is written as A∩ B and read as'A intersection B'. The symbol " ∩" is used to denote "intersection".
EXERCISE -O
1) If A={1,2,3},
B={1,3,5,7},
C={1,3,5,7} then find
(i) A∪B (ii) A∩C (iii) B∩C
2) If A={1,2,3,4,5,6}, B={2,3,4,7,8}
and C= {3,4,5,6} then find
i) A∪B∪C ii) A∩B∩C iii) A∪B∩C
3) If A={1,3,4}, B={2,5,7},C={2,4,6} then prove
a) (A∪B)∪C=A∪(B∪C)
b) (A∩B)∪(A∩C)=A∩(B∪C)
4) Given A={1,2,3}, B={2,4,5},C={1,3}
then Find
i) A∩B ii) A∩C
iii) (A∩B)∪(A∩C) iv) B∪C
v) A∩(B∪C)
Hence, verify
A∩(B∪C)= (A∩B)∪(A∩C)
5) If A, B, C be three subset S when
S={1,2,3,4,5,6,7}
A={1,3,5,6} , B∩C={1,2,6} find
i) (A∪B)∩(A∪C) ii) (A∩B)∪(A∩C)
6) 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∩C v) B∩D
vi) C∪D vii) C∩D
7) If A={a,b,c,d,e}, B={a,c,e,g} and C={b,c,f,g} verify that
i) (A∪B)∩C=(A∩C)∪(B∩C)
(A∩B)∪C= (A∪C)∩(B∪C)
• Difference of Two Sets:
The difference of two sets A and B is the set A - B which contains only elements that are in A but not in B.
EXERCISE-- P
1) Let A= {13,6,12,15, 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
2) Given A= {1, 2, 3, 4} , B={3,4,5}
C={1,4,5} verify
A - (B∪C)= (A -B)∩(A - C)
3) If A={1, 2,3,4} , B={2,3,4,5} , C={1,3,4,5,6,7} find
i) A - B ii) A - C
iii) verify A - (B∩C)= (A-B)∪(A-C)
4) If A={1,2,3,a ,b} and
B={a,b,c,d} find A-B , B-A
• Disjoint Sets:
Two sets A and B are said to be disjoint, if A ∩ B = θ.
• Symmetric Difference of Two Sets:
The symmetric difference of two sets A and B is the set
(A - B)∪(B - A) and is denoted by
A ∆ B. Thus, A ∆ B = (A - B) ∪(B-A)
• Complement of a Set:
The Complement of a set A with respect to the universal set U is the set of all elements of ∪ which are not in A.
It is denoted by A' or Aᶜ or U - A.
Thus A'= { x belongs to U: x ∉ A}
EXERCISE - Q
1) If S={1,2,3,4,5} be the universal set and A={4,5} prove A∪A′ = S
2) If S={0,1,2,3…..8,9} be the universal set A={1,3,5,7}, B{0,1,2,3}
find (a) A′ (b) A′∩B′
(c ) (A∪B)′ ( d) (A-B)′
3) If A={1,2,3},B={2,34,5},
C={3,4,5,6} prove
a) (A∩B)∪(A∩C)=A∩(B∪C)
b) A-(B∪C)=(A-B)∩(A-C)
4) Let the set A and B be given by,
A={1,2,3,4}, B={2,4,6,8,10} and the universal set S={1,2,3,4,5,6,7,7,8,10}
find i) (A∪B)′ ii) (A∩B)ᶜ
5) Let S={1,2,3,4,5} be the universal set and let A={3,4,5} and B={1,4,5}
verify (A∪B)′= A′ ∩B′
6) S={1,2,4,8,16,32} be the universal set and A={1,2,8,32}, B={4,8,32} verify i) (Aᶜ)ᶜ = A
ii) (A∩B)ᶜ=Aᶜ ∪ Bᶜ
iii) (A∪B)ᶜ = Aᶜ ∩ Bᶜ
EXERCISE -- R
1) If S={a,b,c,d,e,f} be the universal set A={a,c,d,f}, (B∩C)={a,b,f}, find i)(A∪B)∩(A∪C) ii) B′∪C′
2) If A={x | -1 ≤ x ≤2} and
B={x| 0 < y ≤4} Find
i) A∪B ii) A∩B
iii) A - B iv) A∪B - (A∩B)
3) Find the set A, B, C if
A∪B = {p,q,r,s}, A∪C={q,r,s,t},
A∩B= {q,r}, A∩C= {q,s}
4) If P={a,b,c,d,e,f} and Q={a,c,e,f}, prove that (P - Q)∪(P∩Q)=P
5) Given A = {1,2,3,4,5} and
B∪C= {3,4,5} find
i) (A∩B)∪(A∩C) ii) (A-B)∩(A-C)
6) Let Z be the set of integers and A={x | x= 6n, n∈ Z},
B={ x |x= 4n, n∈ Z} find A∩B
7) A={x | 2 ≤ x <5}, B={x | 3 < x <7} and iniversal set S={x |0 < x ≤10} verify (A∪B)ᶜ= Aᶜ∩Bᶜ
8) If U={a,b,c,d,e,f} be the universal set and A, B, C are three subset of U, where A={a,c,d} and B∪C={a,d,c,f}, find
i) (A∩B)∪(A∩C) ii) B′∩C′
9) Given X∪Y={1,2,3,4} ,
X ∪ Z={2,3,4,5}, X∩Y={2,3} and
X ∩ Z= {2,4} Find X, Y, Z
10) Let A={1,2,4,5}, B={2,3,5,6} and C={4,5,6,7} verify that
i) A∪(B∩C)=(A∪B)∩(A∪C)
A∩(B∪C)=(A∩B)∪(B∩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)
1) If 65% students know Bengali and 85% students know Hindi, what is the percentage of students who knew both Hindi and Bengali.
2) If 75% boys like apples and 57% boys like oranges, what is the percentage of boys who like both apples and oranges ?
3) During a clinical survey it was observed that 66% patients suffer from cold and 84% patients suffer from fever. What is the percentage of patients who suffer both from cold and fever ?
4) During a survey of 100 students it was observed that 40 studies Mathematics, 52 studies Physics and 35 studies Chemistry; 4 studies all three subjects; 20 studies Math and Physics, 12 studies Math and Chemistry, 16 studies Physics and Chemistry. Find the numbers
(i) Who studied Math only
(ii) Who studied Physics only
(iii) Who studied Chemistry only
(iv) Who studied none of these
5) In a survey of 150 students it was found that 40 students studied Economics, 50 students Mathematics, 60 students studied Accountancy and 15 students studied all the three subjects. It was also found that 27 students studied Economics and Accountancy, 35 students studied Accountancy and Mathematics and 25 students studied Economics and Mathematics. Find the number who studied only Economics and the number who studied none of these subjects ?
6) If a set A has 4 elements and a set B has 6 elements what can be the minimum number of elements in the set AUB ?
7) Out of 1600 students in a school 399 played cricket, 580 played football, 450 played hockey, 90 played both cricket and hockey, 125 played hockey and football and 155 played cricket and football, 50 played all the three games. How many students did not play any game ?
8) In a statically investigation of 1003 families of Calcutta, it was found that 478 families had neither radio nor a T. V, 438 families had a radio and 227 a T. V. How many families in that group has both radio and a T. V?
9) The production manager of Sen., Sarkar & Lahiri Company Examined 100 items produced by the workers and furnished following reports to his boss: Defect in measurement 50, defect in Coloring 30, defect in quality 23, defect in quality and Coloring 10, defect in measurement & Coloring 8, defect in measurement and quality 20 and 5 are defective in all aspects. The manager was penalized for the report. Using appropriate result of set theory. Explain the reason for the penal measure.
10) Using set operations find the
H. C. F of the number
i) 12, 15 ii) 12, 15, 18
iii) 30,105,165. iv) 15, 40 and 105
11) Using set operations find L.C.M of the number
i) 6,42,105. ii) 12, 15, 20
iii) 15, 25, 30
12) If A and B are two sets containing 4 and 7 distinct elements respectively, find th minimum possible number of and maximum possible number of elements A∪B
i) 5,10. ii) 4,12. iii) 7,11. iv) 8,13
13) If A, B, C are three sets in which n(A∩B∩C)= 8, n(A∩B)=15, n(A)=22, n(B∩C)=11, n(B)=19=n(C) , (A∩C)=10, then find A∪B∪C
i) 31. ii) 33. iii) 35. iv) 32
14) Set X, Y and Z are such that
X={A:A²-5A+6=0};
Y={B: B²-8B+15=0} and
Z={C: C²-7C+10=0}. Find
(X∩Y∩Z)∩ (X∪Y∪Z)
a) { } b) {2,5}. c) {2,3,5}. d) {2,5}
15) At a party , 30% of people have only cocktail drinks and snacks, 20% have only food, and 70% have snacks. If nobody has all the three and everybody has something, then how many people (%) have food (given cocktail and snacks are not taken alone).
a) 60%. b) 50%. c) 70%. d) none
16) 2 sets M and N are defined as follows:
M={S,T,R,A,N,D} and
N={S,T,A,N,D,A,R,D}. the above two set are:
A) equal. B) equivalent.C) disjoint. D) none
17) A survey of a society whose residents are mainly gangsters shows that one-sixth of the society residents own bungalows, one-fourth own flats and one-eighth of those who own bungalows do not own flats.
A) what fraction of the society residents owns both a bungalow and a flat?
i) 7/12. ii) 7/48. iii) 7/16. iv) none
B) What fraction of the society residents owns either bungalow or a flat or both ?
i) 4/5. ii) 1/48. iii) 13/48 iv) 15/48
C) What fraction of the society residents owns neither a bungalow nor a flat?
i) 35/48. ii) 15/16. iii) 30/44 iv) n
18) Total students is 2800. English students only 650, Hindi students only 550, Bengali students 450 only, students took all the three subjects 100, students who took Hindi as well as English is 200, students who took Hindi as well as bengali is 400. Bengali as well as English is 300
A) find the number of Bengali students.
I) 950. ii) 1050. iii) 650. iv) 550
B) Find the number of students who didn't take any subject?
i) 450. ii) 2650. iii) 2550. iv) 550
C) Find the number of students take only one subject
i) 450 ii) 1100. iii) 1600. iv) 1650
D) find the number of students taking atleast two subjects.
i) 600. I) 400. iii) 700. iv) 500
E) The ratio of students taking Hindi to that of Bengali is:
i) 2:1 ii) 1:1. iii) 1:2. iv) 3:2
19) In the ISC Board examination last year. 53% passed in Economics, 61% passed in English, 60% in Mathematics, 24% in Economics and English. 35% in English and Mathematics, 27% in Economics and Mathematics and 5% none.
A) % of pass in all subjects
a) nil b) 7. c) 12. d) 10
B) If the number of students in the class is 200, how many passed in only one subject?
a) >50. b) 46. c) <40. d) 48
C) If the number of students in the class is 300, what will be the% change in the number of passes in only two subjects, if the original number of students is 200?
a) >50%. b) <50%. c) 50%. d) none
D) What is the ratio of% of passes in Economics and Mathematics but not English in relation to the% of passes in Mathematics and English but not economics?
a) 5:7. b) 4:5. c) 7:5. d) none
20) In an Institute, it was found that 48% preferred coffee, 54% liked iced tea and 64% smoked. Of the total 28% liked coffee and iced tea, 32% smoked and drank iced tea and 30% smoked and drank coffee. Only 6% did none of these. If the total number of students is 2000 then:
A) The ratio of the number of students who like only coffee to the number who like only iced tea
a) 3:2. b) 2:3. c) 9:8. d) 8:9
B) The number of students who like coffee and smoking but not iced tea
a) 440. b) 240. c) 300. d) 270
C) The % of those who like coffee or iced tea but not smoking those who like atleast one of these?
a) >30%. b) <30%. c) < 25% d) n
D) The % of those who like atleast one of these is:
a) 94. b) nil. c) 91. d) 100
E) The two items having the ratio 1:2 are :
a) Iced tea only and iced tea and smoking only.
b) coffee and smoking only and iced tea only
c) coffee and tea but not smoking and smoking but not coffee and tea.
d) none
F) The number of persons who like coffee and smoking only and the number who like coffee only bear a ratio.
a) 2:3. b) 3:2. c) 2:1. d) 1:2
G) Percentage of those who like iced tea and smoking but not coffee is:
a) 14.9 b) < 14 c) 14 d) more than 14
21) A & B are two sets such that n(A)=17, n(B)=23, n(A∪B)= 38, then (A∩B)= ?
A) 12. B) 2. C) 7. D) 9
22) In a club, all the members participate either in the Kambola or the Kete. 420 participate in the Kete, 350 participate in the Kambola and 220 participate in both. How many members does the club have ?
A) 1550. B) 550. C) 750. D) 960
23) There are 100000 people living in DLF Colony, New Delhi. Out of them 45000 subscribe to Start Gold Network and 60000 to Zoo TV Network. If 20000 subscribe to both, how many do not subscribe to any of the two?
A) 25000. b) 15000 c) 27000 d)25550
24) At the bachelor party of Sheri, 40 persons chose to kiss him and 25 chose to shake hands with him. 10 persons chose to both kiss him and shake hands with him. How many persons turned out at the party ?
A) 45. B) 55. C) 47. D) 53
25) Set X and Y had 6 and 12 elements respectively. What can be the minimum number of elements in (X∪Y) ?
A) 14. B) 16. C) 12. D) 18
26) In a class of 200 students, 110 students have passed in Economics and 134 students have passed in political science. Then the number of students who have passed in political science only is:
A) 66. B) 44. C) 90. D) 20
27) If A= {1,2,3}, B={3,4}, C={4,5,6}
Then (AXB)∩(BXC) = ?
A) { } B) {(3,4)}. C) {(2,3)(3,2)(3,4)}. D)n
28) Let A={O,R,A,N,G,E} and
B={0,1,2,3,4,5}. The above two sets are :
A) equal. B) equivalent
C) disjoint. D) both B and C
29) There are 40 students in DI Class and 60 students in PS Class. Find the number of students which are either DI class or PS Class if the two classes meet at different hours and 20 students are enrolled in both the subjects.
A) 40. B) 20. C) 80. D) 60
30) The number of elements in the power set of a set containing "p" element is:
A) 2ᵖ⁻¹ B) 2ᵖ C) 2ᵖ⁺¹ D) 2ᵖ +1
31) 30 MBAs went to a B-school festival. 25 MBSs choose to irritate M. Techs while 20 chose to irritate M. Scs. How many chose to irritate both M. Techs and M. Scs ?
A) 20. B) 15. C) 5. D) 10
32) In the final of Miss India contest, between two contestants, half the number of judges voted for Miss Kolkata, one-fourth of them voted for Miss Kerala, 2 voted for both, whereas 8 did not vote for any one of them. How many judges were there in the panel?
A) 32 B) 24 C) 20 D) none
33) The number of non empty subsets of the {8,9,10,11,15} is:
A) 32. B) 31. C) 30. D) 33
34) O & M, a market research group Conducted a survey of 20000 consumers and reported that 17200 consumers liked product K and 14500 consumers liked product L. What is the least number that must have liked both the products ?
A) 20000. B) 17200 C) 14500. D) 7000
35) If the set A has M elements and set B had N elements then the number of elements in AxB is :
A) m+n B) mn. C) m - n. D) mⁿ
36) Out of 450 students who appeared at the BA Examination from a centre, 135 failed in A, 150 failed in M and 137 in C. Those who failed in both A and M were 93, in M and C were 98 and in A and C were 106. The number of students who failed in all the three subjects was 75. Assume that each student appeared in all the three subjects.
A) Find the number of students who failed in a least one of the three subjects.
a) 200 b) 210. c) 205. d) 250
B) Find the number of students who passed in all the three subjects.
a) 300. b) 350. c) 250. d) none
37) In a group of 400, each of whom is atleast accountant or management consultant or sales manager, it was found that 160 are only accountant, 220 are only management consultant and 260 are sales manager, 50 are accountant as well as sales manager, 20 are accountant as well as management consultant as well as sales manager. Find the number of those who are accountant as well as management consultant but not sales manager.
A) 45. B) 40. C) 30. D) 25
38) In a college there are 2500 students. It was revealed that 652 students played football, 405 played hockey and 820 played cricket. But only 212 played both football and hockey, 73 played hockey and cricket, and 87 both cricket and football. Also 76 students played all the three games. The number of students who didn't play at all is:
A) 430. B) 431. C) 429. D) 49
39) Set A and B has 3 and 6 elements respectively. What is the minimum of elements in A∪B ?
A) 18. B) 9. C) 6. D) 3
40) In a company X, 5 employees drink tea, coffee and milk, 4 employees drink tea and milk, 1 employee drinks tea and coffee. No employee takes only milk and coffee, 8 employees take only tea, 13 employees take only milk and 9 employees take only coffee. Similarly in company Y, 7 employees drink tea, coffee and milk, 4 employees drink only tea, 1 employee drinks only coffee and 12 employees drink only milk. There are 10 common employees in company X and Y who drink only milk.
i) What are the total number of employees in X and Y together?
A) 83. B) 73. C) 63. D) 93
ii) What is the ratio of number of employees of X taking milk to the number of employees of Y taking milk?
A) 13/12.B) 11/13.C)3/8.D) 22/15
41) A shop has only red, green and blue carpets. 60% of the carpets have red colour, 30% have green colour and 50% have blue colour. If no carpet has all the three colours, what percentage of the carpets have only one colour?
A) 40%. B) 60%. C) 70%. D) none
42) Two finite sets have p and q number of elements. The total number of subsets of the first set is eight times the total number of subsets of the second set. Find the value of p - q.
A) 2. B) 3. C) 4. D) none
43) Out of 1600 students in a college, 390 played Kho-Kho, 450 played Kabaddi, and 500 played cricket; 90 played both Kho-Kho and Kabaddi; 125 played Kabaddi and Cricket, and 155 played Kho-Kho and Cricket; 50 played all the three games.
A) how many students didn't play any game ?
i) 400. B) 500. C) 450. D) none
B) How many played only Kho-Kho ?
i) 295. B) 195. C) 95. D) 1000
C) How many played only one game?
A) 1030. B) 930 C) 750. D) 730
D) How many played only two games ?
i) 220. B) 320. C) 120 D) none
44) 85 children went to an amusement park where they could ride on the marry-go-round, roller coaster and Ferris wheel. It was known that 20 of them took all the three rides and 55 of them took atleast two of the three. Each ride cost Rs 1, and the total receipts of the amusement park was Rs145.
A) how many children didn't try any of the rides ?
i) 5. B) 15. C) 20. D) 10
B) how many children took exactly one ride ?
i) 5. B) 10. C) 15. D) 20
45) A survey of 235 families to find the type of heating used - gas, electricity or oil -- revealed the following-- 20 houses use oil, of which 8 use gas also and 10 use electricity as well as oil and two use both gas and electricity besides oil. In all 175 houses use either gas or electricity and 132 use electricity.
A) number of houses using is :
i) 90. B) 141. C) 41. D) none
B) number of houses using only electricity is :
i) 82. ii) 130. iii) 12. iv) none
C) Amongst the house using one type of heating, percentage, using electricity only is :
i) 46. B) 34. C) 36. D) none
D) number of houses using gas or oil is :
i) 150. ii) 153. iii) 175. D) none
E) Number of houses using two types of fuel, but not all the three is :
i) 50. B) 54. C) 45. D) none
a) A∪B=B∪A
b) (A∪B)∪C=A∪(B∪C)
c) A∩B=B∩A
d) A∩(B∩C)=(A∩B)∩C
e) A∪(B∩C)=(A∪B)∩(A∪C)
f) A∩(B∪C)=(A∩B)∪(A∩C)
g) (A∪B)′=A′∩B′
h) (A∩B)′ = A′ ∩ B′
i) (A-B)∩(A-C)=(A-C) - (B∪C)
j) (A -B)∪(A-C)=A-(B∩C)
k) (A-B)∪A=A
l) (A-B)∩∅
m) (A-B)∪B=A∪B
n) (A-B)∪ B=A iff B is the subset of A
o) (A∩B)∪(A-B)=A
p) A⨯(B∪C)=(A⨯B)∪(A⨯C)
q) A⨯(B∩C)=(A⨯B)∩(A⨯C)
r) (A∪B)⨯C=(A⨯C)∪(B⨯C)
s) (A∩B)⨯C=(A⨯C)∩(B⨯C)
t) (A∩B)⨯(X ∩Y)=(A⨯X)∩(B⨯Y)
u) (A-B)⨯C=(A⨯C)-(B⨯C)
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