**Relations and Functions **is part of Class 11 Maths for Quick Revision. Here we have given Class 11 Maths **Relations and Functions.**

**Ordered Pair**

An ordered pair consists of two objects or elements in a given fixed order.

**Equality of Two Ordered Pairs**

Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d.

**Cartesian Product of Two Sets**

For any two non-empty sets A and B, the set of all ordered pairs (a, b) where a ∈ A and b ∈ B is called the cartesian product of sets A and B and is denoted by A × B.

Thus, A × B = {(a, b) : a ∈ A and b ∈ B}

If A = Φ or B = Φ, then we define A × B = Φ

Note:

- A × B ≠ B × A
- If n(A) = m and n(B) = n, then n(A × B) = mn and n(B × A) = mn
- If atieast one of A and B is infinite, then (A × B) is infinite and (B × A) is infinite.

Contents

**Relations**

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product set A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

The set of all first elements in a relation R is called the domain of the relation B, and the set of all second elements called images is called the range of R.

Note:

- A relation may be represented either by the Roster form or by the set of builder form, or by an arrow diagram which is a visual representation of relations.
- If n(A) = m, n(B) = n, then n(A × B) = mn and the total number of possible relations from set A to set B = 2
^{mn}

**Inverse of Relation**

For any two non-empty sets A and B. Let R be a relation from a set A to a set B. Then, the inverse of relation R, denoted by R^{-1} is a relation from B to A and it is defined by

R^{-1} ={(b, a) : (a, b) ∈ R}

The domain of R = Range of R^{-1} and

Range of R = Domain of R^{-1}.

**Functions**

A relation f from a set A to set B is said to be function, if every element of set A has one and only image in set B.

In other words, a function f is a relation such that no two pairs in the relations have the first element.

**Real-Valued Function**

A function f : A → B is called a real-valued function if B is a subset of R (set of all real numbers). If A and B both are subsets of R, then f is called a real function.

**Some Specific Types of Functions**

Identity function: The function f : R → R defined by f(x) = x for each x ∈ R is called identity function.

Domain of f = R; Range of f = R

**Constant function:**The function f : R → R defined by f(x) = C, x ∈ R, where C is a constant ∈ R, is called a constant function.

Domain of f = R; Range of f = C**Polynomial function:**A real valued function f : R → R defined by f(x) = a_{0}+ a_{1}x + a_{2}x^{2}+…+ a_{n}x^{n}, where n ∈ N and a_{0}, a_{1}, a_{2},…….. a_{n}∈ R for each x ∈ R, is called polynomial function.**Rational function:**These are the real function of type**The modulus function:**The real function f : R → R defined by f(x) = |x|

or

for all values of x ∈ R is called the modulus function.

Domaim of f = R

Range of f = R^{+} U {0} i.e. [0, ∞)

**Signum function:**The real function f : R → R defined

by f(x) = , x ≠ 0 and 0, if x = 0

or

is called the signum function.

Domain of f = R; Range of f = {-1, 0, 1}

**Greatest integer function:**The real function f : R → R defined by f (x) = {x}, x ∈ R assumes that the values of the greatest integer less than or equal to x, is called the greatest integer function.

Domain of f = R; Range of f = Integer**Fractional part function:**The real function f : R → R defined by f(x) = {x}, x ∈ R is called the fractional part function.

f(x) = {x} = x – [x] for all x ∈R

Domain of f = R; Range of f = [0, 1)**Algebra of Real Functions**

Addition of two real functions: Let f : X → R and g : X → R be any two real functions, where X ∈ R. Then, we define (f + g) : X → R by

{f + g) (x) = f(x) + g(x), for all x ∈ X.**Subtraction of a real function from another:**Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, we define (f – g) : X → R by (f – g) (x) = f (x) – g(x), for all x ∈ X.**Multiplication by a scalar:**Let f : X → R be a real function and K be any scalar belonging to R. Then, the product of Kf is function from X to R defined by (Kf)(x) = Kf(x) for all x ∈ X.**Multiplication of two real functions:**Let f : X → R and g : X → R be any two real functions, where X ⊆ R. Then, product of these two functions i.e. f.g : X → R is defined by (fg) x = f(x) . g(x) ∀ x ∈ X.**The quotient of two real functions:**Let f and g be two real functions defined from X → R. The quotient of ‘f’ by g denoted by is a function defined from X → R as

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## Class 11 Maths Notes

Trigonometric Functions

Principle of Mathematical Induction

Complex Numbers and Quadratic Equations

Linear Inequalities

Permutations and Combinations

Binomial Theorem

Sequences and Series

Straight Lines

Conic Sections

Introduction to Three Dimensional Geometry

Limits and Derivatives

Mathematical Reasoning

Statistics

Probability