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Class: | 12 |

Subject: | Maths – गणित |

Chapter 1: | Relations and Functions |

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Table of Contents

- 1 NCERT solutions for class 12 maths chapter 1 Relations and Functions
- 2 NCERT Chapter to study online and answers given in the end of NCERT books.
- 3 These books are very good for revision and more practice. These book are also confined to NCERT Syllabus.
- 4 Assignments for practice
- 5 Mixed Chapter Tests
- 5.1 Determine whether each of the following relation are reflexive, symmetric and transitive: Relation R in the set A = {1, 2, 3… 13, 14} defined as R = {(x, y): 3x – y = 0}
- 5.2 निम्नलिखित फलन की एकैक (Injective) तथा आच्छादि (Surjective) गुणों की जाँच कीजिए: f(x)=x^2 द्वारा प्रदत्त f:N→N फलन है।
- 5.3 Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b^2} is neither reflexive nor symmetric nor transitive.
- 5.4 Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
- 5.5 सिद्ध कीजिए कि f(x)=[x] द्वारा प्रदत्त महत्तम पूर्णांक फलन f:R→R, न तो एकैकी है और न आच्छादक है, जहाँ [x], x से कम या उसके बराबर महत्तम पूर्णांक को निरूपित करता है।
- 5.6 मान लीजिए कि A={1,2,3}, B={4,5,6,7} तथा f={(1,4),(2,5),(3,6)} A से B तक एक फलन है। सिद्ध कीजिए कि f एकैकी है।
- 5.7 Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
- 5.8 Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.
- 5.9 मान लीजिए कि N में *, a * b = a तथा b का H.C.F. द्वारा परिभाषित द्विआधारी संक्रिया है। क्या * क्रमविनिमय है? क्या * साहचर्य है? क्या N में इस द्विआधारी संक्रिया के तत्समक का अस्तित्व है?
- 5.10 Show that the relation R in the set A of points in a plane given by R = {(P, Q): Distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.
- 5.11 Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?
- 5.12 Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
- 5.13 Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.
- 5.14 Show that the function f: R → R given by f(x) = x^3 is injective.

### NCERT solutions for class 12 maths chapter 1 Relations and Functions

#### Solutions of NCERT exercises given in the chapter

- 12 Maths Chapter 1 Exercise 1.1 Solutions
- 12 Maths Chapter 1 Exercise 1.2 Solutions
- 12 Maths Chapter 1 Exercise 1.3 Solutions
- 12 Maths Chapter 1 Exercise 1.4 Solutions
- 12 Maths Chapter 1 Miscellaneous Exercise 1 Solutions

### NCERT Chapter to study online and answers given in the end of NCERT books.

### These books are very good for revision and more practice. These book are also confined to NCERT Syllabus.

### Assignments for practice

**Mixed Chapter Tests**

Chapter 1, 2, 3 & 4

**Level 1 Test 1 **

**Level 2 Test 1 **

**Level 3 Test 1 Test 2 **

*EXPECTED BACKGROUND KNOWLEDGE*

Before studying this lesson, you should know:

- Concept of set, types of sets, operations on sets
- Concept of ordered pair and cartesian product of set.
- Domain, co-domain and range of a relation and a function

**Relation**

Let A and B be two sets. Then a relation R from Set A into Set B is a subset of A × B.

Types of Relations

(i) Reflexive Relation

(ii) Symmetric Relation

(iii) Transitive Relation

**EQUIVALENCE RELATION**

A relation R on a set A is said to be an equivalence relation on A iff

(i) it is reflexive

(ii) it is symmetric

(iii) it is transitive

**CLASSIFICATION OF FUNCTIONS**

Let f be a function from A to B. If every element of the set B is the image of at least one element of the set A i.e. if there is no unpaired element in the set B then we say that the function f maps the set A onto the set B. Otherwise we say that the function maps the set A into the set B.

Functions for which each element of the set A is mapped to a different element of the set B are said to be one-to-one.

A function can map more than one element of the set A to the same element of the set B. Such a type of function is said to be many-to-one. A function which is both one-to-one and onto is said to be a bijective function.

**BINARY OPERATIONS:**

Let A, B be two non-empty sets, then a function from A × A to A is called a binary operation on A.

If a binary operation on A is denoted by *, the unique element of A associated with the ordered pair (a, b) of A × A is denoted by a * b.

The order of the elements is taken into consideration, i.e. the elements associated with the pairs (a, b) and (b, a) may be different i.e. a * b may not be equal to b * a.

Let A be a non-empty set and ‘*’ be an operation on A, then

Table of Contents

- 1 Determine whether each of the following relation are reflexive, symmetric and transitive: Relation R in the set A = {1, 2, 3… 13, 14} defined as R = {(x, y): 3x – y = 0}
- 2 निम्नलिखित फलन की एकैक (Injective) तथा आच्छादि (Surjective) गुणों की जाँच कीजिए: f(x)=x^2 द्वारा प्रदत्त f:N→N फलन है।
- 3 Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b^2} is neither reflexive nor symmetric nor transitive.
- 4 Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
- 5 सिद्ध कीजिए कि f(x)=[x] द्वारा प्रदत्त महत्तम पूर्णांक फलन f:R→R, न तो एकैकी है और न आच्छादक है, जहाँ [x], x से कम या उसके बराबर महत्तम पूर्णांक को निरूपित करता है।
- 6 मान लीजिए कि A={1,2,3}, B={4,5,6,7} तथा f={(1,4),(2,5),(3,6)} A से B तक एक फलन है। सिद्ध कीजिए कि f एकैकी है।
- 7 Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
- 8 Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.
- 9 मान लीजिए कि N में *, a * b = a तथा b का H.C.F. द्वारा परिभाषित द्विआधारी संक्रिया है। क्या * क्रमविनिमय है? क्या * साहचर्य है? क्या N में इस द्विआधारी संक्रिया के तत्समक का अस्तित्व है?
- 10 Show that the relation R in the set A of points in a plane given by R = {(P, Q): Distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.
- 11 Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?
- 12 Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
- 13 Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.
- 14 Show that the function f: R → R given by f(x) = x^3 is injective.

#### Determine whether each of the following relation are reflexive, symmetric and transitive: Relation R in the set A = {1, 2, 3… 13, 14} defined as R = {(x, y): 3x – y = 0}

R = {(x, y): 3x − y = 0}

∴ R = {(1, 3), (2, 6), (3, 9), (4, 12)}

R is not reflexive since (1, 1), (2, 2) … (14, 14) ∉ R.

Also, R is not symmetric as (1, 3) ∈ R, but (3, 1) ∉ R. [3(3) − 1 ≠ 0] Also, R is not transitive as (1, 3), (3, 9) ∈ R, but (1, 9) ∉ R. [3(1) − 9 ≠ 0] Hence, R is neither reflexive, nor symmetric, nor transitive.

#### निम्नलिखित फलन की एकैक (Injective) तथा आच्छादि (Surjective) गुणों की जाँच कीजिए: f(x)=x^2 द्वारा प्रदत्त f:N→N फलन है।

माना, किसी x, y ∈ N के लिए, f(x) = f(y)

⇒ x^2 = y^2

⇒ x = y.

∴ f एकैक है।

यहाँ, 2 ∈ N, लेकिन, N में x का कोई ऐसा मान नहीं है कि f(x) = x2 = 2.

∴ f आच्छादि नहीं है।

अतः, फलन f एकैक है परन्तु आच्छादि नहीं है।

#### Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b^2} is neither reflexive nor symmetric nor transitive.

It can be observed that (1/2,1/2)∉R, since, 1/2>(1/2)^2

∴ R is not reflexive.

Now, (1, 4) ∈ R as 1 < 42 But, 4 is not less than 12. ∴ (4, 1) ∉ R ∴ R is not symmetric. Now, (3, 2), (2, 1.5) ∈ R [as 3 < 22 = 4 and 2 < (1.5)^2 = 2.25] But, 3 > (1.5)^2 = 2.25

∴ (3, 1.5) ∉ R

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

#### Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.

A relation R is defined on set A as: R = {(a, b): b = a + 1}

∴ R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} we can find (a, a) ∉ R, where a ∈ A.

For instance, (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∉ R

∴ R is not reflexive.

It can be observed that (1, 2) ∈ R, but (2, 1) ∉ R.

∴ R is not symmetric.

Now, (1, 2), (2, 3) ∈ R but, (1, 3) ∉ R

∴ R is not transitive

Hence, R is neither reflexive, nor symmetric, nor transitive.

#### सिद्ध कीजिए कि f(x)=[x] द्वारा प्रदत्त महत्तम पूर्णांक फलन f:R→R, न तो एकैकी है और न आच्छादक है, जहाँ [x], x से कम या उसके बराबर महत्तम पूर्णांक को निरूपित करता है।

यहाँ, f(1.2) = [1.2] = 1 और f(1.9) = [1.9] = 1,

इसलिए f(1.2) = f(1.9) लेकिन 1.2 ≠ 1.9.

∴ f एकैकी फलन है।

हम जानते हैं कि सभी दशमलव की संख्याएँ वास्तविक संख्याएँ होती हैं, जैसे 0.7 ∈ R.

यहाँ, 0.7 ∈ R, लेकिन, R में x का कोई ऐसा मान नहीं है कि f(x) = 0.7

अतः, महत्तम पूर्णांक फलन न तो एकैकी है और न आच्छादक है।

#### मान लीजिए कि A={1,2,3}, B={4,5,6,7} तथा f={(1,4),(2,5),(3,6)} A से B तक एक फलन है। सिद्ध कीजिए कि f एकैकी है।

फलन f: A → B इस प्रकार परिभाषित है कि

f = {(1, 4), (2, 5), (3, 6)}.

∴ f (1) = 4,

f (2) = 5,

f (3) = 6

यहाँ A के प्रत्येक अवयव के लिए B में एक अद्वितीय अवयव है।

अतः, फलन f एकैकी फलन है।

#### Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

A relation R on A is defined as R = {(1, 2), (2, 1)}.

It is clear that (1, 1), (2, 2), (3, 3) ∉ R, ∴ R is not reflexive.

Now, as (1, 2) ∈ R and (2, 1) ∈ R, then R is symmetric.

Now, (1, 2) and (2, 1) ∈ R, however, (1, 1) ∉ R, ∴ R is not transitive.

Hence, R is symmetric but neither reflexive nor transitive.

#### Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.

R = {x, y): x and y have the same number of pages}

Now, R is reflexive since (x, x) ∈ R as x and x has the same number of pages.

Let (x, y) ∈ R ⇒ x and y have the same number of pages.

⇒ y and x have the same number of pages.

⇒ (y, x) ∈ R

∴ R is symmetric.

Now, let (x, y) ∈R and (y, z) ∈ R.

⇒ x and y and have the same number of pages and y and z have the same number of pages.

⇒ x and z have the same number of pages.

⇒ (x, z) ∈ R

∴ R is transitive.

Hence, R is an equivalence relation.

#### मान लीजिए कि N में *, a * b = a तथा b का H.C.F. द्वारा परिभाषित द्विआधारी संक्रिया है। क्या * क्रमविनिमय है? क्या * साहचर्य है? क्या N में इस द्विआधारी संक्रिया के तत्समक का अस्तित्व है?

हम जानते हैं कि a और b का H.C.F. = b और a का H.C.F., सभी a, b ∈ N के लिए

∴ a * b = b * a, अतः, संक्रिया * क्रमविनिमय है।

सभी a, b, c ∈ N के लिए, (a * b)* c = (a और b का H.C.F.) * c = a, b और c का H.C.F.

तथा a *(b * c) = a *(b और c का H.C.F.) = a, b और c का H.C.F.

∴ (a * b) * c = a * (b * c), अतः, संक्रिया * साहचर्य है।

अब, कोई अवयव e ∈ N, संक्रिया * में तत्समक होगा यदि a * e = a = e* a, सभी a ∈ N के लिए

लेकिन ये संबंध किसी भी a ∈ N के लिए सत्य नहीं है।

अतः, N में इस द्विआधारी संक्रिया * के तत्समक का अस्तित्व नहीं है।

Clearly, (P, P) ∈ R since the distance of point P from the origin is always the same as the distance of the same point P from the origin.

∴ R is reflexive.

Now, Let (P, Q) ∈ R.

⇒ The distance of point P from the origin is the same as the distance of point Q from the origin.

⇒ The distance of point Q from the origin is the same as the distance of point P from the origin.

⇒ (Q, P) ∈ R, ∴ R is symmetric.

Now, Let (P, Q), (Q, S) ∈ R.

⇒ The distance of points P and Q from the origin is the same and also, the distance of points Q and S from the origin is the same.

⇒ The distance of points P and S from the origin is the same.

⇒ (P, S) ∈ R

∴ R is transitive.

Therefore, R is an equivalence relation.

The set of all points related to P ≠ (0, 0) will be those points whose distance from the origin is the same as the distance of point P from the origin.

In other words, if O (0, 0) is the origin and OP = k, then the set of all points related to P is at a distance of k from the origin.

Hence, this set of points forms a circle with the centre as the origin and this circle passes through point P.

R is reflexive,

Since (P1, P1) ∈ R, as the same polygon has the same number of sides with itself.

Let (P1, P2) ∈ R. ⇒ P1 and P2 have the same number of sides.

⇒ P2 and P1 have the same number of sides. ⇒ (P2, P1) ∈ R, ∴ R is symmetric.

Now, Let (P1, P2), (P2, P3) ∈ R.

⇒ P1 and P2 have the same number of sides.

Also, P2 and P3 have the same number of sides.

⇒ P1 and P3 have the same number of sides. ⇒ (P1, P3) ∈ R

∴ R is transitive.

Hence, R is an equivalence relation.

The elements in A related to the right-angled triangle (T) with sides 3, 4, and 5 are those polygons which have 3 sides (Since T is a polygon with 3 sides).

Hence, the set of all elements in A related to triangle T is the set of all triangles.

R is reflexive as any line L1 is parallel to itself i.e., (L1, L1) ∈ R.

Now, let (L1, L2) ∈ R. ⇒ L1 is parallel to L2 ⇒ L2 is parallel to L1. ⇒ (L2, L1) ∈ R

∴ R is symmetric.

Now, let (L1, L2), (L2, L3) ∈ R. ⇒ L1 is parallel to L2. Also, L2 is parallel to L3. ⇒ L1 is parallel to L3.

∴ R is transitive.

Hence, R is an equivalence relation.

The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x + 4.

Slope of line y = 2x + 4 is m = 2

It is known that parallel lines have the same slopes.

The line parallel to the given line is of the form y = 2x + c, where c ∈ R.

Hence, the set of all lines related to the given line is given by y = 2x + c, where c ∈ R.

#### Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.

f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}.

gof(1) = g[f(1)] = g(2) = 3 [as f(1) = 2 and g(2) = 3] gof(3) = g[f(3)] = g(5) = 1 [as f(3) = 5 and g(5) = 1] gof(4) = g[f(4)] = g(1) = 3 [as f(4) = 1 and g(1) = 3] ∴ gof = {(1, 3), (3, 1), (4, 3)}

#### Show that the function f: R → R given by f(x) = x^3 is injective.

For one – one

Suppose f(x) = f(y), where x, y ∈ R.

⇒ x^3 = y^3 … (1)

Now, we need to show that x = y.

Suppose x ≠ y, their cubes will also not be equal.

⇒ x^3 ≠ y^3

However, this will be a contradiction to (1).

∴ x = y

Hence, f is injective.