CBSE Class 12 Maths Notes Relations and Functions

Written by

Relations and Functions is part of Class 12 Maths Notes for Quick Revision. Here we have given Class 12 Maths Notes Relations and Functions.

Relation – Relations and Functions

A relation R from set X to a set Y is defined as a subset of the cartesian product X × Y. We can also write it as R ⊆ {(x, y) ∈ X × Y : xRy}.

Note: If n(A) = p and n(B) = q from set A to set B, then n(A × B) = pq and number of relations = 2pq.

Types of Relation

Empty Relation: A relation R in a set X, is called an empty relation, if no element of X is related to any element of X,
i.e. R = Φ ⊂ X × X

Universal Relation: A relation R in a set X, is called universal relation, if each element of X is related to every element of X,
i.e. R = X × X

Reflexive Relation: A relation R defined on a set A is said to be reflexive, if
(x, x) ∈ R, ∀ x ∈ A or
xRx, ∀ x ∈ R

Symmetric Relation: A relation R defined on a set A is said to be symmetric, if
(x, y) ∈ R ⇒ (y, x) ∈ R, ∀ x, y ∈ A or
xRy ⇒ yRx, ∀ x, y ∈ R.

Transitive Relation: A relation R defined on a set A is said to be transitive, if
(x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R, ∀ x, y, z ∈ A
or xRy, yRz ⇒ xRz, ∀ x, y,z ∈ R.

Equivalence Relation: A relation R defined on a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

Equivalence Classes: Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into mutually disjoint subsets A, called partitions or sub-divisions of X satisfying

• all elements of Ai are related to each other, for all i.
• no element of Ai is related to any element of Aj, i ≠ j
• Ai ∪ Aj = X and Ai ∩ Aj = 0, i ≠ j. The subsets Ai and Aj are called equivalence classes.

Function – Relations and Functions

Let X and Y be two non-empty sets. A function or mapping f from X into Y written as f : X → Y is a rule by which each element x ∈ X is associated to a unique element y ∈ Y. Then, f is said to be a function from X to Y.
The elements of X are called the domain of f and the elements of Y are called the codomain of f. The image of the element of X is called the range of X which is a subset of Y.
Note: Every function is a relation but every relation is not a function.

Types of Functions

One-one Function or Injective Function: A function f : X → Y is said to be a one-one function, if the images of distinct elements of x under f are distinct, i.e. f(x1) = f(x2 ) ⇔ x1 = x2, ∀ x1, x2 ∈ X
A function which is not one-one, is known as many-one function.

Onto Function or Surjective Function: A function f : X → Y is said to be onto function or a surjective function, if every element of Y is image of some element of set X under f, i.e. for every y ∈ y, there exists an element X in x such that f(x) = y.
In other words, a function is called an onto function, if its range is equal to the codomain.

Bijective or One-one and Onto Function: A function f : X → Y is said to be a bijective function if it is both one-one and onto.

Composition of Functions: Let f : X → Y and g : Y → Z be two functions. Then, composition of functions f and g is a function from X to Z and is denoted by fog and given by (fog) (x) = f[g(x)], ∀ x ∈ X.
Note
(i) In general, fog(x) ≠ gof(x).
(ii) In general, gof is one-one implies that f is one-one and gof is onto implies that g is onto.
(iii) If f : X → Y, g : Y → Z and h : Z → S are functions, then ho(gof) = (hog)of.

Invertible Function: A function f : X → Y is said to be invertible, if there exists a function g : Y → X such that gof = Ix and fog = Iy. The function g is called inverse of function f and is denoted by f-1.
Note
(i) To prove a function invertible, one should prove that, it is both one-one or onto, i.e. bijective.
(ii) If f : X → V and g : Y → Z are two invertible functions, then gof is also invertible with (gof)-1 = f-1og-1

Domain and Range of Some Useful Functions

Binary Operation: A binary operation * on set X is a function * : X × X → X. It is denoted by a * b.

Commutative Binary Operation: A binary operation * on set X is said to be commutative, if a * b = b * a, ∀ a, b ∈ X.

Associative Binary Operation: A binary operation * on set X is said to be associative, if a * (b * c) = (a * b) * c, ∀ a, b, c ∈ X.
Note: For a binary operation, we can neglect the bracket in an associative property. But in the absence of associative property, we cannot neglect the bracket.

Identity Element: An element e ∈ X is said to be the identity element of a binary operation * on set X, if a * e = e * a = a, ∀ a ∈ X. Identity element is unique.
Note: Zero is an identity for the addition operation on R and one is an identity for the multiplication operation on R.

Invertible Element or Inverse: Let * : X × X → X be a binary operation and let e ∈ X be its identity element. An element a ∈ X is said to be invertible with respect to the operation *, if there exists an element b ∈ X such that a * b = b * a = e, ∀ b ∈ X. Element b is called inverse of element a and is denoted by a-1.
Note: Inverse of an element, if it exists, is unique.

Operation Table: When the number of elements in a set is small, then we can express a binary operation on the set through a table, called the operation table.

We hope the given CBSE Class 12 Maths Notes Relations and Functions will help you. If you have any query regarding NCERT Class 12 Maths Notes Relations and Functions, drop a comment below and we will get back to you at the earliest.

Class 12 Macro Economics Notes

Relations and Functions

Inverse Trigonometric Functions

Matrices

Determinants

Continuity and Differentiability

Application of Derivatives

Integrals

Application of Integrals

Differential Equations

Vector Algebra

Three Dimensional Geometry

Linear Programming

Probability