Course Content
CLASS 12 MATHEMATICS
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Chapter 1: Relation and Functions
Chapter 2: Inverse Trignometry Functions
Chapter 3: Matrices
CLASS 12 MATHEMATICS Chapter wise Formula Notes


1
RELATIONS AND FUNCTIONS

Click any topic to expand & view full definition / formula.

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Ordered Pair

A pair of elements listed in a specific order separated by comma and enclosing the pair in parenthesis is called an ordered pair.

For example, (a, b) is an ordered pair with a as the first element and b as the second element.

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Cartesian Product or Cross Product of sets A and B

The set of ordered pairs (a, b) such that a ∈ A, b ∈ B is called the cartesian product of A to B. The set of ordered pairs (b, a) such that a ∈ A, b ∈ B is called the cartesian product of B to A.

It is written as:
\[
\mathsf{A}\times \mathsf{B} = \{(a,b)\colon \mathsf{a}\in \mathsf{A}, b\in \mathsf{B}\}
\]
\[
\mathsf{B}\times \mathsf{A} = \{(b,a)\colon \mathsf{a}\in \mathsf{A}, b\in \mathsf{B}\}
\]

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Number of elements in A × B
If n(A) = p and n(B) = q then n(A × B) = pq

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Relation from Set A to set B
Let A and B be two non-empty sets, then a relation R from set A to set B is a subset of cartesian product A × B. R ⊆ A X B

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Relation on a Set
Let A be a non-empty set. Then, a relation from A to A is called a relation on set A.

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Domain, Range and Codomain of Relation
Let R be a relation from set A to set B, then the set of all the first elements of the ordered pairs in R is called the domain and the set of all the second elements of the ordered pairs in R is called the range of R, i.e., Domain of R = {a : (a, b) ∈ R} and Range of R = {b : (a, b) ∈ R}. The set B is called the codomain of relation R.

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Empty Relation
A relation from set A to set B is said to be empty if no element of A is related to any element of B, and is denoted by ∅. An empty relation is a subset of A × B.

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Universal Relation
A relation from set A to set B is said to be universal if each element of A is related to every element of B. Universal relation U = A × B.

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Trivial Relations
NOTE: Empty relation and Universal relation are said to be trivial relations.

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Identity Relation
A relation R on the set A is an identity relation if and only if R = {(a, a) for each a ∈ A}

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Types of Relations

A relation on a non-empty set A is said to be

(i) Reflexive, if (a, a) ∈ R for all a ∈ A
(ii) Symmetric, if (a, b) ∈ R implies (b, a) ∈ R, for all a, b ∈ A
(iii) Transitive, if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R, for all a, b, c ∈ A

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

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Equivalence Classes

Let R be an equivalence relation on a set A. The set of all those elements of A, which are related to a, where a ∈ A, is said equivalence class determined by a and is denoted by [a].

Given an arbitrary relation R on an arbitrary set A, R divides A into mutually disjoint subsets Ai, called partitions or subdivisions of A, satisfying the conditions:
(i) All elements of Ai are related to each other, for each i
(ii) No element of Aj is related to any element of Ai, for all i ≠ j
(iii) Ai ∩ Aj = ∅, for all i, j

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Function (Mapping)
For any two non-empty sets A and B, a function f from A to B is a rule or mapping which associates each element of set A to a unique element in set B. It is denoted by f : A → B.

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Domain, Codomain and Range of a Function
Let f : A → B then elements of set A are called domain of f and the elements of set B are called codomain of f. The set of all the images obtained in set B corresponding to each element belongs to A under f is called range.

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Types of Functions
One-one (or injective function): A function f : A → B is called a one-one or injective function, if distinct elements of A have distinct images in B.
i.e., for every x1, x2 ∈ A, f(x1) = f(x2) implies x1 = x2Many-one function: A function f : A → B is called a many-one function, if there exist at least two distinct elements in A, whose images are same in B.Onto (or surjective function): A function f : A → B is said to be onto or surjective function, if every element of B is the image of some elements of A under f.
i.e., for every y ∈ B there exists an element x ∈ A such that f(x) = y

Into function: A function f : A → B is an into function, if there exists an element in B which have no preimage in A.

One-one and onto (or bijective function): A function f : A → B is said to be one-one and onto (or bijective function), if f is both one-one and onto.

In the figures, the functions f1 and f2 are one-one and the functions f3 and f4 are many-one. The functions f2 and f3 are onto and the functions f1 and f4 are into.

Types of functions
Types of functions

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Number of Relations from set A to set B
If n(A) = p and n(B) = q then Number of Relations from A to B = 2pq

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Number of Reflexive Relations on a Set
The number of reflexive relations on a set with the ‘n’ number of elements is given by N = 2n(n-1)

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Number of Symmetric Relations on a Set
Number of Symmetric relations for a set having ‘n’ number of elements is given as N = 2n(n+1)/2

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Number of Functions
If a set A has m elements and set B has n elements, then
The total number of functions from A to B = nm

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Number of Surjective Functions (Onto Functions)
If a set A has m elements and set B has n elements, then
The number of onto functions from A to B = nm – nC1(n-1)m + nC2(n-2)m – nC3(n-3)m + … – nCn-1

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Number of Injective Functions (One to One Functions)
If set A has m elements and set B has n elements, n ≥ m, then the number of injective functions or one to one function is given by n! / (n – m)!

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Number of Bijective functions
If there is bijection between two sets A and B, then both sets will have the same number of elements. If n(A) = n(B) = m, then number of bijective functions = m!.

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