Relations and Functions:Class 12 Maths NCERT Chapter 1

Key Features of NCERT Material for Class 12 Maths Chapter 1 – Relations and Functions

In this Chapter 1:Relations and functions we will learn about relations and functions, types of relation, types of functions.In the next Chapter 2:Inverse trignometric functions we will see basic concepts of inverse trigonometric functions.

Quick revision notes

Relation: A relation R from set X to a set Y is characterized as a subset of the cartesian item X × Y. We can likewise compose it as R ⊆ {(x, y) ∈ X × Y : xRy}.

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

Sorts of Relation

Empty Relation: A relation R in a set X, is called an empty relation, if no component of X is identified with any component of X,

for example R = Φ ⊂ X × X

Universal Relation: A relation R in a set X, is called universal relation, if every component of X is identified with each component of X,

for example R = X × X

Reflexive Relation: A relation R characterized on a set An is supposed to be reflexive, if

(x, x) ∈ R, ∀ x ∈ An or

xRx, ∀ x ∈ R

Symmetric Relation: A relation R characterized on a set An is supposed to be symmetric, if

(x, y) ∈ R ⇒ (y, x) ∈ R, ∀ x, y ∈ An or

xRy ⇒ yRx, ∀ x, y ∈ R.

Transitive Relation: A relation R characterized on a set An is supposed to be transitive, if

(x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R, ∀ x, y, z ∈ A

or then again xRy, yRz ⇒ xRz, ∀ x, y,z ∈ R.

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

Equivalence Classes: Given a subjective equivalence relation R in a discretionary set X, R divides X into commonly disjoint subsets A, called parcels or sub-divisions of X fulfilling

• all elements of Ai are related to one another, 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 known as equivalence classes.

Function: Let there be  X and Y ,two non-empty sets. A function or planning f from X into Y composed as f : X → Y is a standard by which every component x ∈ X is related to an interesting component y ∈ Y. At that point, f is supposed to be a function from X to Y.

The components of X are known as the domain of f and the components of Y are known as the codomain of f. The picture of the component of X is known as the range of X which is a subset of Y.

Note: Every function is a relation yet every relation isn’t a function.

(Chapter 1:Relations and functions)

Sorts of Functions

One-one Function or Injective Function: A fun ction f : X → Y is supposed to be a one-one function, if the pictures of particular components of x under f are unmistakable, for example f(x1) = f(x2 ) ⇔ x1 = x2, ∀ x1, x2 ∈ X

A function which isn’t one-one, is known as the same number of one function.

Onto Function or Surjective Function: A function f : X → Y is supposed to be onto function or a surjective function, if each component of Y is picture of some component of set X under f, for example for each y ∈ y, there exists a component X in x with the end goal that f(x) = y.

As such, a function is called an onto function, if its range is equivalent to the codomain.

Bijective or One-one and Onto Function: A function f : X → Y is supposed to be a bijective function on the off chance that it is both one-one and onto.

Composition of Functions: Let f : X → Y and g : Y → Z be any two functions. At that point, composition of functions f and g is a function from X to Z and is meant by fog and given by (fog) (x) = f[g(x)], ∀ x ∈ X.

Note

(I) when all is said in done, fog(x) ≠ gof(x).

(ii) when all is said in done, gof is one-one infers that f is one-one and gof is onto suggests that g is onto.

(iii) If f : X → Y, g : Y → Z and h : Z → S are functions, at that point ho(gof) = (hog)of.

Invertible Function: A function f : X → Y is supposed to be invertible, if there exists a function g : Y → X with the end goal that gof = Ix and fog = Iy. The function g is called inverse of function f and is signified by f-1.

Note

(I) To demonstrate a function invertible, one ought to demonstrate that, it is both one-one or onto, for example bijective.

(ii) If f : X → V and g : Y → Z are two invertible functions, at that point gof is likewise 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 signified by a * b.

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

Associative Binary Operation: A binary operation * on set X is supposed to be associative, if a * (b * c) = (a * b) * c, ∀ a, b, c ∈ X.

Note: For a binary operation, we can disregard the section in an associative property. Yet, without associative property, we can’t disregard the section.

Identity Element: A component e ∈ X is supposed to be the identity component of a binary operation * on set X, if a * e = e * a = a, ∀ a ∈ X. Identity component is remarkable.

Note: Zero is the identity for the addition operation on R and one is the 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 component. A component a ∈ X is supposed to be invertible with respect to the operation *, if there exists a component b ∈ X with the end goal that a * b = b * a = e, ∀ b ∈ X. Component b is considered inverse of component an and is meant by a-1.

Note: Inverse of a component, on the off chance that it exists, is exceptional.

Operation Table: When the quantity of components in a set is little, at that point we can communicate a binary operation on the set through a table, called the operation table.

(Chapter 1:Relations and functions)

October 29, 2020