Sets, Relations and Functions
Sets, Relations and Functions PDF Notes, Important Questions and Synopsis
SYNOPSIS
 Set:
A set is a welldefined collection of objects and it is denoted by capital letters A, B, ..., Z.  Representation of a set: A set can be represented in two forms:
i. Roster form: All the elements are listed and separated by commas inside the { } braces.
ii. Set builder form: It is the mathematical representation of a set where all the members share a common property listed in the { } braces.  Cardinality of a set: Number of elements present in a set.
 Types of sets:
i. Empty set: A set having no element or cardinality zero is an empty set, i.e. { } or Ø.
ii. Finite set: A set having finite number of elements is a finite set, e.g. A = {1, 2, 3}.
iii. Infinite set: A set having infinite number of elements is an infinite set, e.g. B = {1, 2, 3,...}.
iv. Equal sets: Two sets are equal when they share the same elements and have equal cardinality.
v. Subset: A is a subset of B if B contains all the elements of A.
Note: If A is a subset of B and not equal to B, then A becomes the proper subset of B.
vi. Superset: If B contains all the elements of A, then B is the superset of A.
vii. Power set: The power set of any set A is a set of all the subsets of A and it is denoted by P(A). 
Operations on sets:
i. Disjoint sets: When the sets have no common element, they are called disjoint sets.
ii. Intersection of sets: Intersection of two/more sets is a part (set of elements) which is/are common in those sets ( ).
iii. Union of sets: Union of two sets A and B is a set containing all the elements of A as well as B.
Same applies for n number of sets ( ).
iv.
v. Complement of a set: Complement of set A is a set containing all the elements not in A and denoted by A’.
vi. Difference of two sets: Difference of two sets A and B (A⧍B) is a set containing all the elements of A and B which are not common.
vii. Cartesian product of sets: It is a set of ordered pair of elements containing one object from each set.
It is denoted by A × B, where the first object belongs to the first set and the second object belongs to the second set.
Relations
 Relation: A relation R between two sets is a collection of ordered pairs containing one object from each set.
It can also be written as a Cartesian product of two sets, i.e. R = A × B, where all the elements share a common property. 
Types of relations:
i. Reflexive: A relation R is reflexive if ∀ x ∊ X, (x, x) ∊ R.
ii. Symmetric: A relation R is symmetric if (a, b) ∊ R implies (b, a) ∊ R.
iii. Transitive: A relation R is transitive if (a, b) ∊ R and (b, c) ∊ R implies (a, c) ∊ R.
Note: If R is reflexive, symmetric and transitive, then it is an equivalence relation.
iv. Identity: A relation R is an identity if R = {(x, x):x ∊ X}.
Function:
 Function:A function is a relation where each input has a single output.
It is written as f(x), where x is the input.  Domain, Codomain and Range of a function:
Let f be a function from set A to set B, i.e. f: A→B, then A is the domain and B is the codomain of f.
Here, all the inputs belong to A and the outputs belong to B.
Set containing all the outputs is the range of a function which is denoted by f(A) = {f(a): a ϵ A}.
Note: Clearly, f(A) ϵ B.  Realvalued function:
A function with domain and range both being subsets of a set of real numbers. 
Operations on functions:
Let f, g: A→B be two realvalued functions, then
i. (f ± g)(x) = f(x)± g(x)
ii. (f • g)(x) = f(x) • g(x)
iii. _{}, where g(x)≠0
Note: Domain for all the above functions is 
Classification of functions:

One–one function
A function is a one–one function when each element of domain A is connected with a different element of codomain B. It is also called injective function.
i.e. For a function f: A→B if "x, y ∊ A such that f(x) = f(y) ⇒ x = y  Many–one function
When any two or more elements of domain A are connected with a single element of codomain B, then the function is said to be a many–one function.  Onto function
A function f is said to be an onto function if each element of codomain B is connected with the elements of domain A, i.e. if the range is the same as the codomain.
It is also called a surjective function.
i.e. f: A→B, "y ∊ B, ∃ x ∊ A such that y = f(x).  Into function
A function f is said to be an into function if codomain B has at least one element which is not connected with any of the elements of domain A.  Bijective function
A function f is said to be a bijective function if it is one–one and onto.
 Types of functions:
Trigonometric functions
Function
Domain
Range
f(x) = sin x
(∞, +∞)
[−1,1]
f(x) = cos x
(∞, +∞)
[−1,1]
f(x) = tan x
(∞, +∞)
f(x) = cosec x
(∞,1] U [1, +∞)
f(x) = sec x
(∞,1] U [1, +∞)
f(x) = cot x
(∞, +∞)
 Inverse trigonometric functions
sin^{1}x
cos^{1}x
tan^{1}x
cot^{1}x
sec^{1}x
cosec^{1}x
Domain
[−1,1]
[−1,1]
(−∞,∞)
(−∞,∞)
(−∞,−1]U[1, ∞)
(−∞,−1]U[1, ∞)
Range
[0,Π]
(0,Π)

Exponential functions

Logarithmic functions

Absolute value function
x= 
Greatest integer function
y=[x]= 
Signum function

Fractional part function

Even function
i.e. symmetric about the yaxis 
Odd function
i.e. symmetric about the origin
Related Chapters
 Complex Numbers and Quadratic Equations
 Matrices and Determinants
 Permutations and Combinations
 Mathematical Induction
 Binomial Theorem and its Simple Applications
 Sequences and Series
 Limit, Continuity and Differentiability
 Integral Calculus
 Differential Equations
 Coordinate Geometry
 Three Dimensional Geometry
 Vector Algebra
 Statistics and Probability
 Trigonometry
 Mathematical Reasoning