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ETOOS Comprehensive Study Material
For JEE- Main & Advanced
SETS AND RELATION
SETS AND RELATIONS
Set : A set is a collection of well defined objects which are distinct from each other
Set are generally denoted by capital letters A, B, C, .... etc. and the elements of the set by a, b, c .... etc.
If a is an element of a set A, then we write a A and say a belongs to A.
If a does not belong to A then we write a A,
e.g. The collection of first five prime natural numbers is a set containing the elements 2, 3, 5, 7, 11.
Ex. Which of the following are sets? Justify our answer.
(i) The collection of all months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) The collection of all boys in your class.
Sol. (i) The collection of all months of a year beginning with the letter J is a well defined collection of objects because
one can definitely identify a month that belongs to thisv collection. Hence, this collection is a set.
(ii) The collection of ten most talented writers of India is not a well defined collection because the criteria for
determining a writer’s talent may vary from person to person. Hence, this collection is not a set.
(iii) The collection of all boys in your class is a well defined collection because you can definitely identify a boy
who belongs to this collection. Hence, this collection is a set.
Methods to Write a Set
(i) Roster Method or Tabular Method : In this method a set is described by listing elements, separated by
commas and enclose then by curly brackets. Note that while writing the set in roster form, an element is not generally
repeated e.g. the set of letters of word SCHOOL may be written as {S, C, H, O, L}.
Ex. Write the following sets in roster form:
(i) A = {x: x is an integer and –3 < x < 7}.
(ii) B = {x: x is a natural number less than 6}.
Sol. (i) A = {x: x is an integer and –3 < x < 7}
The elements of this set are –2, –1, 0, 1, 2, 3, 4, 5, and 6 only.
Therefore, the given set can be written in roster form as
A = {–2, –1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {x : x is a natural number less than 6}
The elements of this set are 1, 2, 3, 4, and 5 only.
Therefore, the given set can be written in roster form as
B = {1, 2, 3, 4, 5}
(ii) Set builder form (Property Method) : In this we write down a property or rule which gives us all the
element of the set.
A = {x : P(x)} where P(x) is the property by which x A and colon ( : ) stands for ‘such that’
1
MATHS FOR JEE MAIN & ADVANCED
(iv) Symmetric difference of sets : It is denoted by A B and A B = (A – B) (B – A)
(v) Complement of a set : A' = {x : x A but x U} = U – A
e.g. U = {1, 2,........, 10}, A = {1, 2, 3, 4, 5} then A' = {6, 7, 8, 9, 10}
(vi) Disjoint sets : If A B = , then A, B are disjoint
e.g. If A = {1, 2, 3}, B = {7, 8, 9} then A B =
Laws of Algebra of Sets (Properties of Sets)
(i) De-Morgan Laws : (A B)' = A' B' ; (A B)' = A' B'
(ii) A – (B C) = (A – B) (A – C) ; A – (B C) = (A – B) (A – C)
(iii) Distributive Laws : A (B C) = (A B) (A C) ; A (B C) = (A B) (A C)
(iv) Commutative Laws : A B = B A ; A B = B A
(v) Associative Laws : (A B) C = A (B C) ; (A B) C = A (B C)
(vi) Identity law : A U = A; A =A
(vii) Complement law : A A' = U, A A' = , (A')' = A
(viii) Idempotent law : A A = A, A A = A
(ix) A = ; A U = A
A = A ; A U = U
(x) A B A ; A B B
(xi) A A B ; B A B
(xii) A B A B = A
(xiii) A B A B = B
Ex. Let A = {x : x R, |x| < 1] ; B = [x : x R, |x – 1| 1] and A B = R – D, then the set D is-
(A) [x : 1 < x 2] (B) [x : 1 x < 2] (C) [x : 1 x 2] (D) none of these
Sol. A = [x : x R, –1 < x < 1]
B = [ x : x R : x – 1 –1 or x – 1 1]
= [x : x R : x 0 or x 2]
A B = R – D
where D = [x : x R, 1 x < 2]
Thus (B) is the correct answer.
Venn Diagram
U U U U
A B A B A B A B
A–B B–A
Clearly (A – B) (B – A) (A B) = A B
4
MATHS FOR JEE MAIN & ADVANCED
Set : Set is a collection of well defined objects which are distinct from each other. Sets are usually denoted by
capital letters A, B, C,...... and elements are usually denoted by small letters a, b, c,......
If a is an element of a set A, then we write a A and say a belongs to A or a is in A or a is a member of A. If a does
not belongs to A, we write a A.
Standard Notations :
N : A set of natural numbers.
W : A set of whole numbers.
z : A set of integers.
+ –
Z /Z : A set of all positive/negative integers.
Q : A set of all rational numbers.
+ –
Q /Q : A set of all positive/negative rational numbers.
R : A set of real numbers.
+ –
R /R : A set of all positive/negative real numbers.
C : A set of all complex numbers.
Methods for Describing a Set :
(i) Roster/Listing Method/Tabular Form : In this method, a set is described by listing element, separated by
commas, within braces.
e.g. A = {a, e, i, o, u}
(ii) Set Builder/Rule Method : In this method, we write down a property or rule which gives us all the elements of
the set by that rule.
e.g. A = {x : x is a vowel of English alphabets}
Types of Sets
(i) Finite Set : A set containing finite number of elements or no element.
(ii) Cardinal Number of a Finite Set : The number of elements in a given finite set is called cardinal number of
finite set, denoted by n (A).
(iii) Infinite Set : A set containing finite number of elements.
(iv) Empty/Null/Void Set : A set containing no element, it is denoted by or {}.
(v) Singleton Set : A set containing a single element.
(vi) Equal Sets : Two sets A and B are said to be equal, if every element of A is a member of B and every element
of B is a member of A and we write A = B.
(vii) Equivalent Sets : Two sets are said to be equivalent, if they have same number of elements.
If n(A) =n(B), then A and B are equivalent sets.
(viii) Subset and Superset : Let A and B be two sets. If every element of A is an element of B, then A is called
subset of B and B is called superset of A.
Written as A B or B A
(ix) Finite Set : If A is a subset of B and A B, then A is called proper subset of B and we write A B.
(x) Universal Set (U) : A set consisting of all possible elements which occurs under consideration is called a
universal set.
10
SETS AND RELATION
SOLVED EXAMPLES
Ex. 1 Express set A = {x : x N and x = 2n for n N} in roster form.
Sol. A = {2, 4, 6,.........}
Ex. 2 Express set B = {x2 : x < 4, x W} in roster form.
Sol. B = {0, 1, 4, 9}
Ex. 3 Express set A = {2, 5, 10, 17, 26} in set builder form.
Sol. A = {x : x = n2 + 1, nN, 1 n 5}
Ex. 4 The set A = [x : x R, x2 = 16 and 2x = 6] equal-
(A) (B) [14, 3, 4] (C) [3] (D) [4]
2
Sol. x = 16 x = ±4
2x = 6 x = 3
There is no value of x which satisfies both the above equations.
Thus, A =
Hence (A) is the correct answer
Ex. 5 Examine whether the following statements are true or false :
(i) {a, b} / {b, c, a}
(ii) {a, e} / {x : x is a vowel in the English alphabet}
(iii) {1, 2, 3} {1, 3, 5}
(iv) {a} {a, b, c}
Sol. (i) False as {a, b} is subset of {b, c, a}
(ii) True as a, e are vowels
(iii) False as element 2 is not in the set {1, 3, 5}
(iv) False as a {a, b, c} and {a} {a, b, c}
Ex. 6 Two finite sets of have m and n elements respectively the total number of elements in power set of first set is 56 more
thatn the total number of elements in power set of the second set find the value of m and n respectively.
Sol. Number of elements in power set of 1st set = 2m
Number of elements in power set of 2nd set = 2n
Given 2m = 2n + 56
2m – 2n = 56 2n(2m – n – 1) = 23(23 – 1)
n = 3 and m = 6
Ex. 7 Find power set of set A = {1, 2}.
Sol. P(A) = {, {1}, {2}, {1, 2}}
Ex. 8 Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12} then find A B.
Sol. A B = {2, 4, 6, 8, 10, 12}
Ex. 9 Let A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}. Find A – B and B – A.
Sol. A – B = {x : x A and x B} = {1, 3, 5}
Similarly B – A = {8}
17
SETS AND RELATION
Exercise # 1 [Single Correct Choice Type Questions]
PART 1 SETS
1. If A and B are two sets, then A B = A B iff-
(A) A B (B) B A (C) A = B (D) none of these
2. Let A and B be two sets in the universal set. Then A – B equals-
(A) A B' (B) A' B (C) A B (D) none of these
3. Two sets A, B are disjoint iff-
(A) A B = (B) A B (C) A B = (D) A – B = A
4. Which of the following is a null set ?
(A) {0} (B) {x : x > 0 or x < 0}
2
(C) {x : x = 4 or x = 3} (D) {x : x2 + 1 = 0, x R}
5. If A B, then A B is equal to-
(A) A (B) B (C) A' (D) B'
6. If A and B are two sets, then A (A B)' is equal to-
(A) A (B) B (C) (D) none of these
7. If A is any set, then-
(A) A A' = (B) A A' = U (C) A A' = U (D) none of these
8. If A, B be any two sets, then (A B)' is equal to-
(A) A' B' (B) A' B' (C) A B (D) A B
9. If A and B be any two sets, then (A B)' is equal to-
(A) A' B' (B) A' B' (C) A B (D) A B
10. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 5}, B = {6, 7} then A B' is-
(A) B' (B) A (C) A' (D) B.
11. The set of intelligent students in a class is-
(A) a null set (B) a singleton set
(C) a finite set (D) not a well defined collection
12. Which of the following is the empty set
(A) {x : x is a real number and x2 – 1 = 0} (B) {x : x is a real number and x2 + 1 = 0}
(C) {x : x is a real number and x2 – 9 = 0} (D) {x : x is a real number and x2 = x + 2}
13. The set A = {x : x R, x2 = 16 and 2x = 6} is
(A) Null set (B) Singleton set (C) Infinite set (D) None of these
21
MATHS FOR JEE MAIN & ADVANCED
Exercise # 2 [Comprehension Type Questions]
Comprehension # 1
In a group of 1000 people, there are 750 people, who can speak Hindi and 400 people, who can speak Bengali.
1. Number of people who can speak Hindi only is
(A) 300 (B) 400 (C) 500 (D) 600
2. Number of people who can speak Bengali only is
(A) 150 (B) 250 (C) 50 (D) 100
3. Number of people who can speak both Hindi and Bengali is
(A) 50 (B) 100 (C) 150 (D) 200
Comprehension # 2
Let R be a relation defined as R = { (x, y) : y = |x – 1|, x Z and | x | 3}
4. Relation R is equal to :
(A) {(1, 0), (1, 2), (3, 2), (4, 3)} (B) {(–3, 4), (–2, 3), (–1, 2), (0, 1), (1, 0), (2, 1), (3, 2)}
(C) {(4, –3), (3, –2), (2 –1), (1, 0), (2, 3)} (D) None of these
5. Domain of R is :
(A) {0, 1, 2, 3, 4} (B) {1, 3, 4} (C) {– 3, – 2, – 1, 0, 1, 2, 3} (D) {0, 1, 2, 3, 4}
6. Range of R is
(A) {0, 1, 2, 3, 4} (B) {–3, –2, –1, 0, 1, 2, 3} (C) {–4, –3, –1, –2, 0} (D) {–1, 0, 1, 2, 3, 4}
28
SETS AND RELATION
Exercise # 3 [Subjective Type Questions]
1. Write the set of all vowels in English alphabet which precede letter O.
2. Classify the following as a finite or infinite set :
(i) A = {x N : (x – 1) (x – 2) = 0} (ii) B = {x N : x is odd}
3. Write the following set by roster method : The set of all natural numbers ‘x’ such that 4x + 9 < 50.
4. Describe the following set by set property method {0, 3, 8, 15, 24, 35}
5. Describe the following set by roster method the set of all letters in the word TRIGONOMETRY.
6. Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total
number of subsets of the second set. Find the values of m and n.
7. Which of the following are true ?
(i) If A = {1, 5, 5, 5}, B = {1, 3, 5}, then A B.
(ii) If A = {x : x3 – 1 = 0, x N}, B = {x : x2 – 4x + 3 = 0, x N} then A B.
8. Assume that P(A) = P(B). Prove that A = B.
9. If A = {x : x = 4n + 1, n 5, n N} and B {3n : n 8, n N}, then find A – (A – B).
10. Prove that A B = A B iff A = B.
11. Prove that : A – (B C) = (A – B) (A – C) without using venn diagram.
12. Prove by using venn diagram
(i) A – (B C) = (A – B) (A – C) (ii) A B B A
13. A and B are two sets such that n(A) = 3 and n(B) = 6.
Find (i) minimum value of n(A B) (ii) maximum value of n(A B)
14. Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in
the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and
cricket. Eight play all the three games. Find the total number of members in the three athletic teams.
15. 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. Find the number of students who have taken exactly one subject.
16. Determine the domain and the range of the relation R defined by R = {(x + 1, x + 5) : x {0, 1, 2, 3, 4, 5}}.
17. If A = {3, 4, 6}, B = {1, 3} and C = {1, 2, 6} then find (A – B) × (A – C).
18. Let n be a fixed positive integer. Define a relation R on the set of integers Z, aRb n|(a – b). Then prove that R is
equivalence relation
19. Let R be a relation over the set N × N and it is defined by (a, b) R (c, d) a + d = b + c. Then prove that R is
equivalence relation
20. Let L be the set of all straight lines in the Euclidean plane. Two lines 1 and 2 are said to be related by the relation
R if 1 is parallel to 2. Then prove that R is equivalence relation.
21. For n, m N, n | m means that n is a factor of m, then prove that relation | is reflexive, transitive but not symmetric.
22. Let R = {(x, y) : x, y A, x + y = 5} where A = {1, 2, 3, 4, 5} then prove that R is neither reflexive nor transitive but
symmetric.
29
MATHS FOR JEE MAIN & ADVANCED
Exercise # 4 Part # I [Previous Year Questions] [AIEEE/JEE-MAIN]
1. Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is-
(A) transitive (B) not symmetric (C) reflexive (D) a function [AIEEE-2004]
2. Let R = {(3, 3), (6, 6), (9, 9), (12, 12) (6, 12), (3, 9), (3, 12), (3, 6)} be relation on the set A = {3, 6, 9, 12}. Then the
relation R is [AIEEE - 2005]
(A) reflexive and transitive only (B) reflexive only
(C) an equilvalence relation (D) reflexive and symmetric only
3. Let W denote the words in the english dictionary. Define the relation R by : R = {(x, y) W × W | the words x and y
have at least one letter in common}. Then R is- [AIEEE - 2006]
(A) reflexive, symmetric and not transitive (B) reflexive, symmetric and transitive
(C) reflexive, not symmetric and transitive (D) not reflexive, symmetric and transitive
4 The set S : {1, 2, 3, ...., 12} is to be partitioned into three sets A, B, C of equalsize.Thus A B C = S,
A B = B C = A C = . The number of ways to partition S is- [AIEEE - 2007]
3 4 3 4
(A) 12!/3!(4!) (B) 12!/3!(3!) (C) 12!/(4!) (D) 12!/(3!)
5. Let R be the real line. Consider the following subsets of the plane R × R [AIEEE-2008]
S = {(x, y) : y = x + 1 and 0 < x < 2}
T = {(x, y) : x – y is an integer}
Which one of the following is true ?
(A) T is an equivalence relation on R but S is not (B) Neither S nor T is an equivalence relation on R
(C) Both S and T are equivalence relations on R (D) S is an equivalence relation on R but T is not
6. If A, B and C are three sets such that A B = A C and A B = A C, then [AIEEE-2009]
(A) A = C (B) B = C (C) A B = (D) A = B
7. Consider the following relations : [AIEEE-2010]
R : {(x, y)| x ,y are real numbers and x = wy for some rational number w}
m p
S = { n , q | m, n, p and q are integers such that n, q 0 and qm = pn}
Then
(A) neither R nor S is an equivalence relation
(B) S is an equivalence relation but R is not an equivalence relation
(C) R and S both are equivalence relations
(D) R is an equivalence relation but S is not an equivalence relation
8. Let R be the set of real numbers. [AIEEE-2011]
Statement-1 : A = {(x, y) R × R : y – x is an integer} is an equivalence relation on R.
Statement-2 : B = {(x, y) R × R : x = y for some rational number } is an equivalence relation on R.
(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(B) Statement-1 is true, Statement-2 is true;Statement-2 is not a correct explanation for Statement-1.
(C) Statement-1 is true, Statement-2 is false. (D) Statement-1 is false, Statement-2 is true.
30
11th Class Modules Chapter Details
Physics Chemistry Mathematics
5 5 5
Modules Modules Modules
PHYSICS CHEMISTRY MATHEMATICS
Module-1 Module-1(PC) Module-1
1. Physical World and Units 1. Mole Concept 1. Basic Maths and Logarithm
& Dimensions 2. Atomic Structure 2. Quadratic Equation
2. Basic Maths & Vector 3. Chemical Bonding 3. Sequence and Series
3. Kinematics 4. Gaseous State
Module-2
Module-2 Module-2(PC) 1. Trigonometric Ratio and
1. Thermodynamics Identities
1. Newton’s Law of Motion 2. Trigonometric Equation
& Friction 2. Thermochemistry
3. Chemical Equilibrium 3. Properties & Solution
2. Work, Energy & Power of Triangle
4. Ionic Equilibrium
Module-3 Module-3
Module-3(IC)
1. Centre of Mass & Collisions 1. Periodic Table & Its Properties 1. Permutation & Combination
2. Rotational Motion 2. Redox Reaction & Equivalent 2. Binomial Theorum
3. Gravitation Concepts 3. Complex Number
3. Hydrogen & Its Components
Module-4 4. S-Block Module-4
1. Straight Line
1. Mechanical Properties 2. Circle
of Matter Module-4(OC)
1. Nomenclature of 3. Conic Section
2. Thermal Properties of Matter (Parabola,Ellipse & Hyperbola)
Organic Compounds
Module-5 2. Isomerism
3. General Organic Chemistry Module-5
1. Simple Harmonic Motion 1. Mathematical Induction
2. Wave Motion Module-5(OC) 2. Mathematical Reasoning
3. Measurement Error 1. Reaction Mechanism 3. Statistics
& Experiment 2. Hydrocarbon
3. Aromatic Hydrocarbon
4. Environmental Chemistry
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12th Class Modules Chapter Details
Physics Chemistry Mathematics
5 5 5
Modules Modules Modules
PHYSICS CHEMISTRY MATHEMATICS
Module-1 Module-1(PC) Module-1
1. Solid State 1. Sets & Relation
1. Electrostatics
2. Solutions and 2. Function
2. Capacitance
Colligative Properties 3. Inverse Trigonometric Function
3. Electro Chemistry 4. Probability
Module-2
1. Current Electricity Module-2(PC) Module-2
2. Magnetic Effect of Current 1. Chemical Kinetics and
and Magnetism Nuclear Chemistry 1. Limit
2. Surface Chemistry 2. Continuity
Module-3 3. Differentiability
Module-3(IC) 4. Method of Differentiation
1. Electromagnetic Induction
2. Alternating Current 1. Metallurgy
2. P- Block Module-3
Module-4 3. Transition Elements 1. Indefinite Integration
(d & f block) 2. Definite Integration
1. Geometrical Optics 4. Co-ordination Compound 3. Area Under the Curve
2. Wave Optics 5. Salt Analysis & Qualitative
Analysis Module-4
Module-5
Module-4(OC) 1. Application of Derivative
1. Modern Physics
1. Alkyl Halides & Aryl Halides 2. Matrix
2. Nuclear Physics
2. Alcohol, Phenol & Ether 3. Determinant
3. Solids & Semiconductor
Devices 3. Carbonyl Compound
Module-5
4. Electromagnetic Waves
5. Principle of Communication Module-5(OC) 1. Differential Equation
1. Carboxylic Acid & Their 2. Vector & 3-Dimensional
Derivatives
2. Biomolecules & Polymers
3. Chemistry in Everyday Life
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