1 RELATIONS AND FUNCTIONS

KEY CONCEPT INVOLVED

1. Relations - Let A and B be two non-empty sets then every subset of A × B defines a relation from A to B
and every relation from A to B is a subset of A × B.
Let R  A × B and (a, b)  R. then we say that a is related to b by the relation R as aRb. If (a, b)  R as
a R b.
2. Domain and Range of a Relation - Let R be a relation from A to B, that is, let R  A × B. then Domain
R = {a : a A, (a, b) R for some b B} i.e. dom. R is the set of all the first elements of the ordered pairs
which belong to R. Range R = (b : b B, (a, b) R for some a A} i.e. range R is the set of all the second
elements of the ordered pairs which belong to R. Thus Dom. R  A, Range R  B.
3. Inverse Relation - Let R  A × B be a relation from A to B. Then inverse relation R–1  B × A is defined by
R–1 {(b, a) : (a, b) R}
It is clear that
(i) aRb = bR–1 a
(ii) dom. R–1 = range R and range R–1 = dom R.
(iii) (R–1)–1 = R.
4. Composition of Relation - Let R A × B, S B × C be two relations. Then composition of the relations
R and S is denoted by SoR A × C and is defined by (a, c)  (SoR) iff b  B such that (a, b) 
R, (b, c) S.
5. Relations in a set - let A () be a set and R A × A i.e. R is a relation in the set A.
6. Reflexive Relations - R is a reflexive relation if (a, a) R,  a R it should be noted that if for any a A
such that a R a. then R is not reflexive.
7. Symmetric Relation - R is called symmetric relation on A if (x, y) R (y, x) R.
i.e. if x is related to y, then y is also related to x.
It should be noted that R is symmetric iff R–1 = R.
8. Anti Symmetric Relations - R is called an anti symmetric relation if (a, b) R and (b, a) R a = b.
Thus if a  b then a may be related to b or b may be related to a but never both.
9. Transitive Relations - R is called a transitive relation if (a, b) R (b, c) R (a, c) R
10. Identity Relations - R is an identity relation if (a, b) R iff a = b. i.e. every element of A is related to only
itself and always identity relation is reflexive symmetric and transitive.
11. Equivalence Relations - a relation R in a set A is called an equivalence relation if
(i) R is reflexive i.e. (a, a) R  a A
(ii) R is symmetric i.e. (a, b) R (b, a) R
(iii) R is transitive i.e. (a, b), (b, c) R (a, c) R.
12. Functions - Suppose that to each element in a set A there is assigned, by some rule, an unique element of
a set B. Such rules are called functions. If we let f denote these rules, then we write f : A  B as f is a
function of A into B.
13. Equal Functions - If f and g are functions defined on the same domain A and if f (a) = g (a) for every
a A, then f = g.
14. Constant Functions - Let f : A  B. If f (a) = b, a constant, for all a A, then f is called a constant function.
Thus f is called a constant function if range f consists of only one element.
15. Identity Functions - A function f is such that A  A is called an identity function if f (x) = x,  x A it is
denoted by IA.
16. One-One Functions (Injective) - Let f : A  B then f is called a one-one function. If no two different
elements in A have the same image i.e. different elements in A have different elements in B.
Denoted by symbol f is one-one if
f (a) = f (a)  a = a
i.e. a  a f (a)  f (a)
A mapping which is not one-one is called many one function.
17. Onto functions (Surjective) - In the mapping f : A  B, if every member of B appears as the image of
atleast one element of A, then we say “f is a function of A onto B or simply f is an onto functions” Thus
f is onto iff f (A) = B
i.e. range = codomain
A function which is not onto is called into function.
18. Inverse of a function - Let f : A  B and b B then the inverse of b i.e. f–1 (b) consists of those elements
in A which are mapped onto b i.e. f–1 (b) = {x ; x A, f (x) b}
 f–1 (b) A, f–1 (b) may be a null set or a singleton.
19. Inverse Functions - Let f : A  B be a one-one onto-function from A onto B. Then for each b B.
f–1 (b) A and is unique. So, f–1 : B  A is a function defined by f–1 (b) = a, iff f (a) = b.
Then f–1 is called the inverse function of f. If f has inverse function, f is also called invertible or non-
singular.
Thus f is invertible (non-singular) iff it is one-one onto (bijective) function.
20. Composition Functions - Let f : A  B and g : B  C, be two functions,
Then composition of f and g denoted by gof : A  C is defined by (gof) (a) = g {f (a)}.
21. Binary Operation - A binary operation  on a set A is a function  : A × A  A. We denote  (a, b) by a  b
22. Commutative Binary Operation - A binary operation  on the set A is commutative if for every a, b A,
a  b = b  a.
23. Associative Binary Operation - A binary operation  on the set A is associative if
(a  b)  c = a  (b  c).
24. An Identity Element e for Binary Operation - Let  : A × A  A be a binary operation. There exists an
element e  A such that a  e = a = e  a  a A, then e is called an identity element for Binary Operation  .
25. Inverse of an Element a - Let  : A × A  A be a binary operation with identity element e in A.
an element a A is invertible w.r.t. binary operation  , if there exists an element b in A such that
a  b = e = b  a. and b is called the inverse of a and is denoted by a–1.

CONNECTING CONCEPTS
1. In general gof  fog.
2. f : A  B, be one-one, onto then
f–1 of = IA and fof–1 = IB
3. f : A  B, g : B  C, h : C  D
then (hog) of = ho (gof).
4. f : A  B, g : B  C be one-one and onto then gof : A  C is also one-one onto and (gof)–1 = f–1 o g–1.
5. Let : A  B, then IB of = f and foIA = f. It should be noted that foIB is not defined since for
(foIB) (x) = fo {IB (x)} = f (x)
IB (x) exist when x B and f (x) exist when x A
6. f : A  B, g : B  C are both one-one, then gof : A  C is also one-one it should be noted that for gof to
be one-one f must be one-one.
7. If f : A  B g : B  C are both onto then gof must be onto. However, the converse is not true. But for gof
to be onto g must be onto.
8. The domain of the functions
(f + g) (x) = f (x) + g (x)
(f – g) (x) = f (x) – g (x)
(fg) (x) = f (x) g (x)
f (x)
is given by (dom. f)  (dom g) while domain of the function (f/g) (x) = is given by..
g (x)
(dom f) (dom. g) – {x : g (x) = 0}
9. If O (A) = m, O (B) = n, then total number of mappings from A to B is nm.
10. If A and B are finite sets and O (A) = m, O (B) = n, m  n.
n!
Then number of injection (one-one) from A to B is nPm = (n  m)!
11. If f : A  B is injective (one-one), then O(A)  O (B).
12. If f : A  B is surjective (onto), then O (A)  O (B).
13. If f : A  B is bijective (one-one onto), then O (A) = O (B).
14. Let f : A  B and O (A) = O (B), then f is one-one  it is onto.
15. Let f : A  B and X1, X2  A, then f is one-one iff f (X1  X2) = f (X1) f (X2)
16. Let f : A  B and X A, Y B, then in general f–1 (f (x)) X, f (f–1 (y)) Y
If f is one-one onto f–1 (f (x)) = x, f (f–1 (y)) = Y.
Class 12 Maths NCERT Solutions
NCERT Solutions Important Questions NCERT Exemplar
Chapter 1 Relations and Chapter 1 Relations and
Relations and Functions
Functions Functions
Chapter 2 Inverse Chapter 2 Inverse
Concept of Relations and Functions
Trigonometric Functions Trigonometric Functions
Chapter 3 Matrices Binary Operations Chapter 3 Matrices
Chapter 4 Determinants Inverse Trigonometric Functions Chapter 4 Determinants
Chapter 5 Continuity and Chapter 5 Continuity and
Matrices
Differentiability Differentiability
Chapter 6 Application of Chapter 6 Application of
Matrix and Operations of Matrices
Derivatives Derivatives
Transpose of a Matrix and Symmetric
Chapter 7 Integrals
Chapter 7 Integrals Ex 7.1 Matrix
Inverse of a Matrix by Elementary Chapter 8 Applications of
Integrals Class 12 Ex 7.2 Operations Integrals
Chapter 9 Differential
Determinants
Integrals Class 12 Ex 7.3 Equations
Integrals Class 12 Ex 7.4 Expansion of Determinants Chapter 10 Vector Algebra
Chapter 11 Three Dimensional
Properties of Determinants
Integrals Class 12 Ex 7.5 Geometry
Inverse of a Matrix and Application of Chapter 12 Linear
Integrals Class 12 Ex 7.6 Determinants and Matrix Programming
Integrals Class 12 Ex 7.7 Continuity and Differentiability Chapter 13 Probability
Integrals Class 12 Ex 7.8 Continuity
Integrals Class 12 Ex 7.9 Differentiability
Integrals Class 12 Ex 7.10 Application of Derivatives
Rate Measure Approximations and
Integrals Class 12 Ex 7.11 Increasing-Decreasing Functions
Integrals Class 12
Tangents and Normals
Miscellaneous Exercise
Chapter 8 Application of
Maxima and Minima
Integrals
Chapter 9 Differential
Integrals
Equations
Chapter 10 Vector Algebra Types of Integrals
Chapter 11 Three Dimensional
Differential Equation
Geometry
Chapter 12 Linear
Formation of Differential Equations
Programming
Chapter 13 Probability Ex Solution of Different Types of Differential
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13.1 Equations
Probability Solutions Ex 13.2 Vector Algebra
Probability Solutions Ex 13.3 Algebra of Vectors
Probability Solutions Ex 13.4 Dot and Cross Products of Two Vectors
Probability Solutions Ex 13.5 Three Dimensional Geometry
Direction Cosines and Lines
Plane
Linear Programming
Probability
Conditional Probability and Independent
Events
Baye’s Theorem and Probability
Distribution

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Chapter 12: Higher Order

Chapter 1: Relations Chapter 23 Algebra of Vectors
Derivatives
Chapter 13: Derivative as a Rate Chapter 24: Scalar Or Dot
Chapter 2: Functions
Measurer Product
Chapter 14: Differentials, Errors Chapter 25: Vector or Cross
Chapter 3: Binary Operations
and Approximations Product
Chapter 4: Inverse Trigonometric
Chapter 15: Mean Value Theorems Chapter 26: Scalar Triple Product
Functions
Chapter 27: Direction Cosines
Chapter 5: Algebra of Matrices Chapter 16: Tangents and Normals
and Direction Ratios
Chapter 17: Increasing and
Chapter 6: Determinants Chapter 28 Straight line in space
Decreasing Functions
Chapter 7: Adjoint and Inverse of a
Chapter 18: Maxima and Minima Chapter 29: The plane
Matrix
Chapter 8: Solution of
Chapter 19: Indefinite Integrals Chapter 30: Linear programming
Simultaneous Linear Equations
Chapter 9: Continuity Chapter 20: Definite Integrals Chapter 31: Probability
Chapter 21: Areas of Bounded Chapter 32: Mean and variance of
Chapter 10: Differentiability
Regions a random variable
Chapter 11: Differentiation Chapter 22: Differential Equations Chapter 33: Binomial Distribution

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JEE Main Maths Chapter wise Previous Year Questions

1. Relations, Functions and Reasoning

2. Complex Numbers
3. Quadratic Equations And Expressions
4. Matrices, Determinatnts and Solutions of Linear Equations
5. Permutations and Combinations
6. Binomial Theorem and Mathematical Induction
7. Sequences and Series
8. Limits,Continuity,Differentiability and Differentiation
9. Applications of Derivatives
10. Indefinite and Definite Integrals
11. Differential Equations and Areas
12. Cartesian System and Straight Lines
13. Circles and System of Circles
14. Conic Sections
15. Three Dimensional Geometry
16. Vectors
17. Statistics and Probability
18. Trignometry
19. Miscellaneous

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