Relations and Functions ML Aggarwal ISC Class-12 Understanding APC Mathematics Solutions Chapter-1. Step by step Solutions of ML Aggarwal ISC Understanding APC Mathematics Class-12  Exercise 1, Exe 2, Exe 3, Exe 4, Exe 5, Exe 6  With Chapter Test Questions. Visit official Website for detail information about ISC Board Class-12 Mathematics.

## Relations and Functions ML Aggarwal ISC Class-12 Maths ch-1

 Class: 12th Subject: Mathematics Chapter  : Ch-1 Relations and Functions Page-18
 Board ISC Writer ML Aggarwal ISC Understanding Vol-I Publications APC Arya Publications

-: Select Topics :-

Exe-1.1,

Exe-1.2,

Exe-1.3,

Exe-1.4,

Exe-1.5,

Exe-1.6,

Chapter Test

### Relations and Functions ML Aggarwal ISC Class-12 Maths ch-1

Relation

The concept of relation is used in relating two objects or quantities with each other. Suppose two sets are considered, the relationship between them will be established if there is a connection between the elements of two or more non-empty sets.

Mathematically, “a relation R from a set A to a set B is a subset of the cartesian product A × B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A × B”.

Types of Relations

A relation R from A to A is also stated as a relation on A, and it can be said that the relation in a set A is a subset of A × A. Thus, the empty set φ and A × A are two extreme relations. Below are the definitions of types of relations:

Empty Relation

If no element of A is related to any element of A, i.e. R = φ ⊂ A × A, then the relation R in a set A is called empty relation.

Universal Relation

If each element of A is related to every element of A, i.e. R = A × A, then the relation R in set A is said to be universal relation.

Both the empty relation and the universal relation are some times called trivial relations.

A relation R in a set A is called-

Reflexive- if (a, a) ∈ R, for every a ∈ A,

Symmetric- if (a1, a2) ∈ R implies that (a2, a1) ∈ R , for all a1, a2∈ A,

Transitive- if (a1, a2) ∈ R and (a2, a3) ∈ R  implies that (a1, a3) ∈ R  for all a1, a2, a3 ∈ A.

Equivalence Relation- A relation R in a set A is an equivalence relation if R is reflexive, symmetric and transitive.

#### Functions

A function is a relationship which explains that there should be only one output for each input. It is a special kind of relation(a set of ordered pairs) which obeys a rule, i.e. every y-value should be connected to only one y-value.

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

In other words, a function f is a relation from a set A to set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element. Also, A and B are two non-empty sets.

### Composition of Functions

Let f: A → B and g: B → C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof: A → C given by;

gof (x) = g(f (x)), ∀ x ∈ A

#### Invertible Functions

A function f : X → Y is defined to be invertible if there exists a function g : Y → X such that gof = IX and fog = IY. The function g is called the inverse of f and is denoted by f–1.

An important note is that, if f is invertible, then f must be one-one and onto and conversely if f is one-one and onto, then f must be invertible.

Binary Operations

A binary operation ∗ on a set A is a function ∗ : A × A → A. We denote ∗ (a, b) by a ∗ b.

Exe-1.1,

### Relations and Functions ML Aggarwal ISC Class-12 Maths ch-1

Page-18, 19, 20

Question 1:

Determine whether each of the following relation are  reflexive symmetric and transitive:

(i) ……………….

(ii) ………………

Question 2:

If the relation ………………… the relation R is :

(i) ……………….

(ii) ………………

Question 3:

…………………..

…………………..

……………………

Question 30:

If R be the relation in the set ………………………….. then determine whether

(i) ……………….

(ii) ………………

(iii) ………………

### Exe-1.2,

Relations and Functions ML Aggarwal ISC Class-12 Maths ch-1

Page-30, 31

Question 1:

A function is defined by ……………………..

Question 2:

Question 3:

…………………….

…………………….

……………………..

Question 24:

Find the range of following function

(i)……………….

(ii)……………….

Exe-1.3,

### Relations and Functions ML Aggarwal ISC Class-12 Maths ch-1

Page-44, 45, 46

Question 1:

Let A = {1, 20}, B ………………………. show that f = g.

Question 2:

Check the injectivity and surjectivity of the following function .

……………..

………………

Question 3:

……………………

…………………….

…………………..

Question 32:

Which of the following graph represent a one-one function.

………………………

### Exe-1.4,

Relations and Functions ML Aggarwal ISC Class-12 Maths ch-1

Page-56, 57

Question 1:

If f = ………………………. find gof.

Question 2:

Let A = { ………………………. then find gof : A–> C.

Question 3:

………………………

………………………..

…………………….

Question 14:

If f(x) = x + 7 …………………………….. (7) .

### Exe-1.5,

Relations and Functions ML Aggarwal ISC Class-12 Maths ch-1

Page-71, 72

Question 1:

Let f : {1, 2, 3} ………………………….. Also show that (f………………….)

Question 2:

Let s = ………………. if it exist

………………

……………….

Question 3:

……………………….

……………………….

………………………

Question 20:

If R : – ………………………………………. Then find f -1.

Exe-1.6,

### Relations and Functions ML Aggarwal ISC Class-12 Maths ch-1

Page-88, 89

Question 1:

For each binary operation ‘*’ defined below, determine whether * is commutative and whether * is associative :

………………

………………..

Question 2:

…………………………

………………………….

………………………..

Question 24:

Let A = Q x Q, where Q is the set of rational number, and ‘*’ be a binary operation …………………… but associative.

### Chapter Test

Relations and Functions ML Aggarwal ISC Class-12 Maths ch-1

Page-90

Question 1:

If a rational R on Z (set of integers) is defined by R = …………………………… but not transitive.

Question 2:

Show that the number of ………………….. (2, 1) is two.

Question 3:

…………………..

……………………

…………………….

Question 14:

Let S be the set of rational number ……………………….. prove that.

(i)……………….

(ii)………………….

-: End of Relations and Functions ML Aggarwal Vol-I ISC Class-12 Solutions:-

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