NCERT Solutions for Chapter 1 Relations and Functions Class 12 Maths is now available on MSVgo. It covers extensive topics under the chapter “Relations and Functions”, enabling students to gain a deeper understanding of certain fundamental topics. Detailed, step-by-step solutions are provided for the exercise questions, which will come in handy if the student is stuck at any time while solving a problem. Solving questions from the Class 12 Maths NCERT Solutions ensures that the students’ learning is aligned with the curriculum, therefore preparing them well for the exams. Going through the Class 12 Maths Chapter 1 Solutions not only aids in the thorough understanding of the chapter, but also boosts the students’ confidence in their knowledge of the subject matter.
Topics Covered in Relations and Functions for CBSE Class 12:
1. Types of Relations
2. Types of Functions
3. Composition of Functions and Invertible Function
4. Binary Operations

You mighthave come across **Relation and Functions** while studying algebra. In mathematics, both these terms have different meanings. So it is paramount first to comprehend the definition of both terms before moving ahead.

When the elements of two or more non-empty sets are related to each other in some way, it is called a relation. There are eight types of relations in mathematics:

__Empty Relation:__ An empty relation exists if the elements of two sets M and N are not connected to each other in any way. It is also called a void relation, and can be written as R* *=* *φ. For example, whenever set M = {4, 5, 6} at that point, one of the empty relations can be R = {m, n} where, |m – n| = 7.

**Empty Relation: R = φ ⊂ M × M**

__Universal Relation:__ When every element in set G is mapped to every element in set H, it is known as a universal relation. For example, if set G = {g, h, i}, one of the universal relations will be R = {g, h} where, |g – h| ≥ 0.

**Universal Relation: R = G × G**

__Identity Relation:__ An identity relation exists if every element of a set is mapped to only itself. For example, in a set L = {j, k, l}, the identity relation will be I = {j, j}, {k, k}, {l, l}.

**Identity Relation: I = {(j, j), j ∈ L}**

__Inverse Relation:__ When every element in set P is the inverse of every element in set Q, it is known as an inverse relation. For example if set P = {(p, q), (r, s)}, then the inverse relation will be R-1 = {(q, p), (s, r)}.

**Inverse Relation = R-1 = {(q, p): (p, q) ∈ R}**

__Reflexive Relation:__ When every element of a set is mapped to itself, it is known as a reflexive relation. For example, in a set Y = {45, 46}, the reflexive relation will be R = {(45, 45), (46, 46), (45, 46), (46, 45)}.

**Reflexive Relation: (y, y) ∈ R**

__Symmetric Relation:__ If p=q is true and q=p is also true, then a symmetric relation exists between the elements of a set. In other words, a relation R is symmetric only if (q, p) ∈ R is true when (p, q) ∈ R. For example, in a set P = {5, 8}, the symmetric relation will be R = {(5, 8), (8, 5)}.

**Symmetric Relation: pRq ⇒ qRp, ∀ p, q ∈ P**

__Transitive Relation:__ A relation is transitive if (j, k) ∈ R and (k, l) ∈ R, then (j, l) ∈ R.

**Transitive Relation: jRk and kRl ⇒ jRl ∀ j, k, l ∈ R**

__Equivalence Relation:__ A relation that is simultaneously reflexive, symmetric and transitive is known as an equivalence relation.

A function between two sets exists if the relation is such that each element of one set is related to exactly one element in the other set. Let X and Y be any two non-empty sets. Mapping from X to Y will be a function only when every element in set X has only one image in set Y. Additionally, a function cannot have two pairs of the same first element. A function from set X to set Y is represented by F: X→Y. There are four types of functions in mathematics.

__One-To-One Function:__A function f: X → Y is one-to-one if each element of set X corresponds to a unique element in set Y. It is also known as an injective function. For example, if x1 ∈ X and x2 ∈ Y, then f is defined as f: X → Y such that f (x1) = f (x2).__Many-To-One Function:__If two or more elements of set X are mapped to the same element of set Y, it is known as a many-to-one function.__Onto Function:__If for every element of set Y, there is at least one or more than one element mapped in set X, then the function is known as onto function. Onto is also referred to as Surjective Function.__One-To-One Correspondence Function:__A one-to-one correspondence function is when the function satisfies both the injective and surjective properties. In other words, the function f maps each element of X with a unique element of Y, and every element of Y has a pre-image in X.

**Composition of Functions**

The composition of a function is an operation where two functions, m and n, generate a new function, o, in such a way that o(x) = m(n(x))[1] [2] . Here, function n is applied to the function of x. In other words, one function is applied to the result of another function.

Let m : P → Q and n : Q → R be two functions. Then the composition of m and n, denoted by n*o*m, is defined as the function n*o*m : P → R given by mΟn(x) = m(n(x)), ∀ x ∈ P.

For example, if m(x) = 3x + 6, and n(x) = x – 4, then m(n(x)) [3] [4] = 3(x – 4) + 6.

Properties of Composite Functions:

- Composite functions are associative, that is (m[5] [6] Οn)[7] [8] Οh = m[9] [10] Ο(n[11] [12] Οh)[13] [14] .
- Composite functions are not commutative, that is m[15] [16] Οn ≠ n[17] [18] Οm.[19] [20]
- The function composition of one-to-one function is always one to one.
- The function composition of two onto functions is always onto.

**Invertible Function**

An inverse function is when a function reverses into another function. In other words, if a function f takes x to y then, the inverse of f will take y to x. If the function is denoted by f or F, then the inverse function is denoted by f-1 or F-1.

For example, if f(x) = x + 3 = y, then g(y) = y – 3 = x, which is f-1(x).

A binary operation ∗ on a set P is a function ∗ : P × P → P.

∗ (p, q) is denoted by p ∗ q.

Some properties of binary operations:

__Commutative Binary Operation:__A binary operation * on set P is commutative if p * q = q * p, ∀ p, q ∈ P.__Associative Binary Operation:__A binary operation * on set M is associative, if m * (n * o) = (m * n) * o, ∀ m, n, o ∈ M.

**Note:** For a binary operation, the bracket in an associative property can be ignored. But in the absence of associative property, the bracket cannot be ignored.

__Identity Element:__An element e ∈ M is the identity element of a binary operation * on set M, if m * e = e * m = m, ∀ m ∈ M. The 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 * : S × S → S be a binary operation and let e ∈ S be its identity element. An element s ∈ S is invertible with respect to the operation *, if there exists an element t ∈ S such that s * t = t * s = e, ∀ t ∈ S. Element t is called the inverse of element s and is denoted by s-1.

**Note:** Inverse of an element, if it exists, is unique.

**Example 1**

Let M be the set of all students of a girls school. Show that the relation R in M given by R = {(m, n) : m is brother [1] [2] of n} is the empty relation and R′ = {(m, n) : the difference between heights of m and n is less than 4 meters} is the universal relation.

**Solution**

Since the school is a girls school, no student of the school can be brother [3] [4] of any student of the school. Hence, R = φ, showing that R is the empty relation. It is also obvious that the difference between heights of any two students of the school has to be less than 4 meters. This shows that R′ = M × M is the universal relation.

**Example 2**

Let P be the set of all triangles in a plane with R a relation in P given by R = {(P1 , P2 ) : P1 is congruent to P2}. Show that R is an equivalence relation.

**Solution**

Because every triangle is congruent to itself, R is reflexive. Moreover, (P1 , P2) ∈ R ⇒ P1 is congruent to P2 ⇒ P2 is congruent to P1 ⇒ (P2, P1) ∈ R. Therefore, R is symmetric. Furthermore, (P1, P2), (P2, P3) ∈ R ⇒ P1 is congruent to P2 and P2 is congruent to P3 ⇒ P1 is congruent to P3 ⇒ (P1, P3) ∈ R. Hence, R is an equivalence relation.

**Example 3**

Show that the relation R in the set {14, 15, 16} given by R = {(14, 14), (15, 15), (16, 16), (14, 15), (15, 16)} is reflexive but neither symmetric nor transitive.

**Solution**

R is reflexive since (14, 14), (15, 15) and (16, 16) lie in R. Additionally, R is not symmetric, as (14, 15) ∈ R but (15, 14) ∉ R. Similarly, R is not transitive, as (14, 15) ∈ R and (15, 16) ∈ R but (14, 16) ∉ R.

**Example 4**

Show that an onto function g : {21, 22, 23} → {21, 22, 23} is always one-one.

**Solution**

Suppose g is not one-one. Then there exists two elements, say 21 and 22 in the domain whose image in the co-domain is the same. Also, the image of 23 under g can be only one element. Therefore, a maximum of two elements of the co-domain {21, 22, 23} can be contained within the range set. This shows that g is not onto, a contradiction. Hence, g must be one-one.

**Example 5**

Show that a one-one function h : {6, 7, 8} → {6, 7, 8} must be onto.

**Solution**

Since h is one-one, three elements of {6, 7, 8} must be taken to three different elements of the co-domain {6, 7, 8} under h. Hence, h has to be onto.

**Example 6**

Consider f : P → N, g : Q → N and h : R → N defined as f(p) = 2p, g(q) = 3q + 4 and h(r) = sin r, ∀ p, q and r in N. Show that h[5] [6] Ο(g[7] [8] Οf) = (h[9] [10] Οg) of.[11] [12]

**Solution**

We have hΟ(gΟf) (p) = h(gΟf (p)) = h(g(f(p))) = h(g(2p)) = h(3(2p) + 4) = h(6p + 4) = sin (6p+4), ∀ p ∈ P.

Also, ((hΟg)Ο f ) (p) =(hΟg) (f(p)) = (hΟg) (2p) = h (g(2p)) = h(3(2p) + 4) = h(6p + 4) = sin (6p + 4), ∀ p ∈ N.

This proves that hΟ(gΟf) = (hΟg) Ο f. This result is true in general situations as well.

**Example 7**

Show that subtraction, addition and multiplication are binary operations on R, but division is not a binary operation on R. Also, show that division is a binary operation on the set R∗ of nonzero real numbers.

**Solution**

+ : R × R → R is given by

(m, n) → m + n

– : R × R → R is given by

(m, n) → m – n

× : R × R → R is given by

(m, n) → mn

Since ‘+’, ‘–’ and ‘×’ are functions, they are binary operations on R.

But ÷: R × R → R, given by (m, n) → m/n, is not a function, therefore not a binary operation, as for n = 0, m/n is not defined.

However, ÷ : R∗ × R∗ → R∗, given by (m, n) → m/n is a function, therefore a binary operation on R∗.

**Example 8**

Show that –q is not the inverse of q ∈ N for the addition operation + on N and 1/q is not the inverse of q ∈ N for multiplication operation × on N, for q ≠ 1.[13] [14]

**Solution**

Since –q ∉ N, –q can not be inverse of q for addition operation on N, although –q satisfies q + (– q) = 0 = (–q) + q.

Similarly, for q ≠ 1 in N, 1/q ∉ N, which implies that other than 1 no element of N has inverse for multiplication operation on N.

**Example 9**

Show that the number of binary operations on {1, 2} having 1 as identity and having 2 as the inverse of 2 is exactly one.

**Solution**

A binary operation ∗ on {1, 2} is a function from {1, 2} × {1, 2} to {1, 2}, i.e., a function from {(1, 1), (1, 2), (2, 1), (2, 2)} → {2, 1}. Since 1 is the identity for the desired binary operation ∗, ∗ (1, 1) = 1, ∗ (1, 2) = 2, ∗ (2, 1) = 2 and the only choice left is for the pair (2, 2). Since 2 is the inverse of 1, i.e., ∗ (2, 2) must be equal to 1. Thus, the number of desired binary operations is only one.

Going through the points in this chapter in detail will help in gaining in-depth knowledge of the lesson, which in turn will help in scoring marks in the exam. Some of the key features of the Class 12 Maths NCERT Solutions are:

- The solutions aid in the understanding of the chapter due to the easy-to-follow explanations provided.
- The NCERT exercises are solved by experts with step-by-step solutions.
- All the solutions are in line with the latest CBSE Syllabus and guidelines.
- Practising the Relations and Functions Class 12 problems ensures the student has strong retention of knowledge of the lesson.

**Can you give a summary of the topics covered in CBSE Class 12 Maths Chapter 1? **

The topic covered in the CBSE Maths Chapter 1 is Relations and Functions Class 12. The sub-topics covered are Types of Relations, Types of Functions, Composition of Functions and Invertible Functions, and Binary Operations. The solutions to the exercise problems are solved by highly experienced professionals. By providing the NCERT Solutions, the aim is to help students understand the lesson and score high marks in the Class 12 CBSE exams.

**Why is MSVgo NCERT Solutions for Class 12 Maths Chapter 1 reliable?**

MSVgo NCERT Solutions for Class 12 Maths Chapter 1 is reliable because the solutions are detailed and have been prepared by experts. Using the MSVgo NCERT Solutions will be helpful while preparing for exams as there are numerous exam-style questions and exercises available for rigorous practice. Watching MSVgo videos on the chapter enhances learning and boosts confidence in the topic.

**How to study for Relations and Functions Class 12 chapter?**

Here are some tips for studying CBSE Relations and Functions Class 12 chapter:

- Watch videos explaining the Relations and Functions Class 12 chapter.
- Read and understand textbook definitions thoroughly.
- Memorize important formulas given in the NCERT Solutions.
- Problems and exercises given in the NCERT Solutions are in accordance with the CBSE board. Practice these exercises to get used to answering board exam-style questions, preparing you well for the exam.

**What videos should I refer to for CBSE class 12 maths chapter 1?**

The videos listed below can be accessed on the MSVgo app. These engaging videos explain the chapter in detail, and will help give a clear understanding of the lessons. Download the app and start watching these videos:

- Equivalence Relations and Classes
- Composition of Functions
- Invertible Functions
- Types of Relations
- Introduction to Functions
- Types of Functions
- Binary Operations

**How is the NCERT Solutions for class 12 maths beneficial to me?**

Here are some reasons that make the NCERT Solutions a useful tool while studying for class 12 maths exams:

- The answers to the problems are detailed and easy to understand.
- The solutions are provided by subject matter experts, making this a reliable source for practice.
- The solutions guide has a list of all the important formulas required in the chapter.
- The solutions help students understand the Class 12 CBSE exam question patterns, marks distribution and more.

Here it is to be noted that all functions are relations, but not all relations are functions. **Relations And Functions **is one of the most important parts of algebra. It is an easy concept that demands conceptual clarity. You can check the MSVgo app to know more about the topic. The MSVgo philosophy is to enable a core understanding of any concept. MSVgo is a video library that explains concepts with examples or explanatory visualizations or animations. Check out videos on MSVgo to understand the concept behind **Relations And Functions. **Happy learning!

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