# Relations and Functions ML Aggarwal ISC Class-12 Maths

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 **CISCE** 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 :-

### 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 (a_{1}, a_{2}) ∈ R implies that (a_{2}, a_{1}) ∈ R , for all a_{1}, a_{2}∈ A,

**Transitive- **if (a_{1}, a_{2}) ∈ R and (a_{2}, a_{3}) ∈ R implies that (a_{1}, a_{3}) ∈ R for all a_{1}, a_{2}, a_{3} ∈ 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 = I_{X} and fog = I_{Y}. 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:**

Is the relation f defined by……….. justify your answer.

**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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