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# JEE Maths Sets, Relations and Functions

## Sets, Relations and Functions PDF Notes, Important Questions and Synopsis

SYNOPSIS

1. Set:
A set is a well-defined collection of objects and it is denoted by capital letters A, B, ..., Z.
2. 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.
3. Cardinality of a set: Number of elements present in a set.

4. 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).
5.  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

1. 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.
2.  Types of relations:
i. Reflexive: A relation R is reflexive if  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:

1. Function:A function is a relation where each input has a single output.
It is written as f(x), where x is the input.
2. Domain, Co-domain 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 co-domain 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.
3. Real-valued function:
A function with domain and range both being subsets of a set of real numbers.
4. Operations on functions:
Let f, g: A→B be two real-valued 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

5.  Classification of functions:

1. One–one function

A function is a one–one function when each element of domain A is connected with a different element of co-domain 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
2. Many–one function
When any two or more elements of domain A are connected with a single element of co-domain B, then the function is said to be a many–one function.
3. Onto function
A function f is said to be an onto function if each element of co-domain B is connected with the elements of domain A, i.e. if the range is the same as the co-domain.
It is also called a surjective function.
i.e. f: A→B, " B,  x  A such that y = f(x).

4. Into function
A function f is said to be an into function if co-domain B has at least one element which is not connected with any of the elements of domain A.

5. Bijective function
A function f is said to be a bijective function if it is one–one and onto.
1. 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 (-∞, +∞)

2. Inverse trigonometric functions

 sin-1x cos-1x tan-1x cot-1x sec-1x cosec-1x Domain [−1,1] [−1,1] (−∞,∞) (−∞,∞) (−∞,−1]U[1, ∞) (−∞,−1]U[1, ∞) Range [0,Π] (0,Π)

3. Exponential functions

4. Logarithmic functions

5.  Absolute value function
|x|=

6. Greatest integer function
y=[x]=

7. Signum function

8. Fractional part function

9. Even function

10. Odd function