1. संबंध एवं पफलन

Source

There is no permanent place in the world for ugly mathematics … . It may be very hard to define mathematical beauty but that is just as true of beauty of any kind, we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognising one when we read it.
दुनिया में बदसूरत गणित के लिए कोई स्थायी जगह नहीं है… गणितीय सुंदरता को परिभाषित करना बहुत कठिन हो सकता है लेकिन यह किसी भी तरह की सुंदरता के लिए उतना ही सच है, हम शायद यह नहीं जानते कि एक सुंदर कविता से हमारा क्या मतलब है, लेकिन जब हम इसे पढ़ते हैं तो यह हमें पहचानने से नहीं रोकता है। [गूगल द्वारा अनुवादित]

G. H. Hardy

1.1 भूमिका (Introduction)

स्मरण कीजिए कि कक्षा XI में, संबंध एवं फलन, प्रांत, सहप्रांत तथा परिसर आदि की अवधारणाओं का, विभिन्न प्रकार के वास्तविक मानीय फलनों और उनके आलेखों सहित परिचय कराया जा चुका है। गणित में शब्द ‘संबंध (Relation)’ की सवंफ़ल्पना को अंग्रेजी भाषा में इस शब्द के अर्थ से लिया गया है, जिसके अनुसार दो वस्तुएँ परस्पर संबंधित होती है, यदि उनके बीच एक अभिज्ञेय (Recognisable) कड़ी हो। मान लीजिए कि $A$ , किसी स्कूल की कक्षा XII के विद्यार्थियों का समुच्चय है तथा $B$ उसी स्कूल की कक्षा XI के विद्यार्थियों का समुच्चय हैं। अब समुच्चय $A$ से समुच्चय $B$ तक के संबंधों के कुछ उदाहरण इस प्रकार हैं

(i) $\{(a, b) ∈ A \times B: a, b$ का भाई है $\}$ ,
(ii) $\{(a, b) ∈ A \times B: a, b$ की बहन है $\}$ ,
(iii) $\{(a, b) ∈ A \times B: a$ की आयु $b$ की आयु से अधिक है $\}$ ,
(iv) $\{(a, b) ∈ A \times B:$ पिछली अंतिम परीक्षा में $a$ द्वारा प्राप्त पूर्णांक $b$ द्वारा प्राप्त पूर्णांक से कम है $\}$ ,
(v) $\{(a, b) ∈ A \times B: a$ उसी जगह रहता है जहाँ $b$ रहता है $\}$ . तथापि $A$ से $B$ तक के किसी संबंध $R$ को अमूर्तरूप (Abstracting) से हम गणित में $A \times B$ के एक स्वेच्छ (Arbitrary) उपसमुच्चय की तरह परिभाषित करते हैं।

यदि $(a, b) ∈ R$ , तो हम कहते हैं कि संबंध $R$ के अंतर्गत $a, b$ से संबंधित है और हम इसे $a R b$ लिखते हैं। सामान्यत:, यदि $(a, b) ∈ R$ , तो हम इस बात की चिंता नहीं करते हैं कि $a$ तथा $b$ के बीच कोई अभिज्ञेय कड़ी है अथवा नहीं है। जैसा कि कक्षा XI में देख चुके हैं, फलन एक विशेष प्रकार के संबंध होता हैं।

इस अध्याय में, हम विभिन्न प्रकार के संबंधों एवं फलनों, फलनों के संयोजन (composition), व्युत्क्रमणीय (Invertible) फलनों और द्विआधारी संक्रियाओं का अध्ययन करेंगे।

1.2 संबंधों के प्रकार (Types of Relations)

इस अनुच्छेद में हम विभिन्न प्रकार के संबंधों का अध्ययन करेंगे। हमें ज्ञात है कि किसी समुच्चय $A$ में संबंध, $A \times A$ का एक उपसमुच्चय होता है। अत: रिक्त समुच्चय $φ ⊂ A \times A$ तथा $A \times A$ स्वयं, दो अन्त्य संबंध हैं। स्पष्टीकरण हेतु, $R = \{(a, b): a - b = 10\}$ द्वारा प्रदत्त समुच्चय A = 4 पर परिभाषित एक संबंध R पर विचार कीजिए। यह एक रिक्त समुच्चय है, क्योंकि ऐसा कोई भी युग्म (pair) नहीं है जो प्रतिबंध a - b = 10 को संतुष्ट करता है। इसी प्रकार $R' = \{(a, b) : | a - b | ≥ 0\}$ , संपूर्ण समुच्चय $A \times A$ के तुल्य है, क्योंकि $A \times A$ के सभी युग्म $(a, b), | a - b | ≥ 0$ को संतुष्ट करते हैं। यह दोनों अन्त्य के उदाहरण हमें निम्नलिखित परिभाषाओं के लिए प्रेरित करते हैं।

परिभाषा 1

समुच्चय $A$ पर परिभाषित संबंध $R$ एक रिक्त संबंध कहलाता है, यदि $A$ का कोई भी अवयव $A$ के किसी भी अवयव से संबंधित नहीं है, अर्थात् $R = φ ⊂ A \times A$ .

परिभाषा 2

समुच्चय $A$ पर परिभाषित संबंध $R$ , एक सार्वत्रिक (universal) संबंध कहलाता है, यदि $A$ का प्रत्येक अवयव $A$ के सभी अवयवों से संबंधित है, अर्थात् $R = A \times A$ .

रिक्त संबंध तथा सार्वत्रिक संबंध को कभी-कभी तुच्छ (trivial) संबंध भी कहते हैं।

उदाहरण 1

मान लीजिए कि $A$ किसी बालकों के स्कूल के सभी विद्यार्थियों का समुच्चय है। दर्शाइए कि $R = \{(a, b) : a, b$ की बहन है $\}$ द्वारा प्रदत्त संबंध एक रिक्त संबंध है तथा $R' = \{(a, b) : a$ तथा $b$ की ऊँचाईयों का अंतर 3 मीटर से कम है $\}$ द्वारा प्रदत्त संबंध एक सार्वत्रिक संबंध है।

हल

प्रश्नानुसार, क्योंकि स्कूल बालकों का है, अतएव स्कूल का कोई भी विद्यार्थी, स्कूल के किसी भी विद्यार्थी की बहन नहीं हो सकता है। अत: $R = φ$ , जिससे प्रदर्शित होता है कि R रिक्त संबंध है। यह भी स्पष्ट है कि किन्हीं भी दो विद्यार्थियों की ऊँचाइयों का अंतर 3 मीटर से कम होना ही चाहिए। इससे प्रकट होता है कि $R' = A \times A$ सार्वत्रिक संबंध है।

कक्षा XI में विद्यार्थीगण सीख चुके हैं कि किसी संबंध को दो प्रकार से निरूपित किया जा सकता है, नामत: रोस्टर विधि तथा समुच्चय निर्माण विधि। तथापि बहुत से लेखकों द्वारा समुच्चय $\{1, 2, 3, 4\}$ पर परिभाषित संबंध $R = \{(a, b) : b = a + 1\}$ को $a R b$ द्वारा भी निरूपित किया जाता है, यदि और केवल यदि b = a + 1 हो। जब कभी सुविधाजनक होगा, हम भी इस संकेतन (notation) का प्रयोग करेंगे।

यदि $(a, b) ∈ R$ , तो हम कहते हैं कि $a,b$ से संबंधित है’ और इस बात को हम $a R b$ द्वारा प्रकट करते हैं।

एक अत्यन्त महत्वपूर्ण संबंध, जिसकी गणित में एक सार्थक (significant) भूमिका है, तुल्यता संबंध (Equivalence Relation) कहलाता है। तुल्यता संबंध का अध्ययन करने के लिए हम पहले तीन प्रकार के संबंधों, नामत: स्वतुल्य (Reflexive), सममित (Symmetric) तथा संक्रामक (Transitive) संबंधों पर विचार करते हैं।

परिभाषा 3

समुच्चय $A$ पर परिभाषित संबंध $R$ ;

(i) स्वतुल्य (reflexive) कहलाता है, यदि प्रत्येक $a ∈ A$ के लिए $(a, a) ∈ R$ ,
(ii) सममित (symmetric) कहलाता है, यदि समस्त $a1, a2 ∈ A$ के लिए $(a1, a2) ∈ R$ से $(a2, a1) ∈ R$ प्राप्त हो।
(iii) संक्रामक (transitive) कहलाता है, यदि समस्त, $a1, a2, a3 ∈ A$ के लिए $(a1, a2) ∈ R$ तथा $(a2, a3) ∈ R$ से $(a1, a3) ∈ R$ प्राप्त हो।

परिभाषा 4

$A$ पर परिभाषित संबंध $R$ एक तुल्यता संबंध कहलाता है, यदि $R$ स्वतुल्य, सममित तथा संक्रामक है।

उदाहरण 2

Let T be the set of all triangles in a plane with $R$ a relation in $T$ given by $R = \{(T1, T2) : T1$ is congruent to $T2\}$ . Show that $R$ is an equivalence relation.

हल

$R$ is reflexive, since every triangle is congruent to itself.

Further, $(T_1, T_2) ∈ R ⇒ T_1$ is congruent to $T_2 ⇒ T_1$ is congruent to $T_1 ⇒ (T_2, T_1) ∈ R$ . Hence, $R$ is symmetric.

Moreover, $(T_1, T_2), (T_2, T_1) ∈ R ⇒ T$ is congruent to $T_2$ and $T_2$ is congruent to $T_3 ⇒ T_1$ is congruent to $T_3 ⇒ (T_1, T_3) ∈ R$ . Therefore, $R$ is an equivalence relation.

उदाहरण 3

Let $L$ be the set of all lines in a plane and R be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1$ is perpendicular to $L_2\}$ . Show that $R$ is symmetric but neither reflexive nor transitive.

हल

$R$ is not reflexive, as a line $L_1$ can not be perpendicular to itself, i.e., $(L_1, L_1) ∉ R$ . $R$ is symmetric as $(L_1, L_2) ∈ R$

$⇒ L_1$ is perpendicular to $L_2$

$⇒ L_2$ is perpendicular to $L_1$

$⇒ (L2, L1) ∈ R$ .

$R$ is not transitive. Indeed, if $L_1$ is perpendicular to $L_2$ and $L_2$ is perpendicular to $L_3$ , then $L_1$ can never be perpendicular to $L_3$ . In fact, $L_1$ is parallel to $L_3$ , i.e., $(L_1, L_2) ∈ R, (L_2, L_3) ∈ R$ but $(L_1, L_3) ∉ R$ .

उदाहरण 4

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

हल

$R$ is reflexive, since $(1, 1)$ , $(2, 2)$ and $(3, 3)$ lie in $R$ . Also, $R$ is not symmetric, as $(1, 2) ∈ R$ but $(2, 1) ∉ R$ . Similarly, $R$ is not transitive, as $(1, 2) ∈ R$ and $(2, 3) ∈ R$ but $(1, 3) ∉ R$ .

उदाहरण 5

Show that the relation R in the set $ℤ$ of integers given by

$R = \{(a, b) : 2$ divides $a - b\}$

is an equivalence relation.

हल

$R$ is reflexive, as $2$ divides $(a - a)$ for all $a ∈ ℤ$ . Further, if $(a, b) ∈ R$ , then $2$ divides $a - b$ . Therefore, $2$ divides $b - a$ . Hence, $(b, a) ∈ R$ , which shows that $R$ is symmetric. Similarly, if $(a, b) ∈ R$ and $(b, c) ∈ R$ , then $a - b$ and $b - c$ are divisible by $2$ . Now, $a - c = (a - b) + (b - c)$ is even (Why?). So, $(a - c)$ is divisible by $2$ . This shows that $R$ is transitive. Thus, $R$ is an equivalence relation in $ℤ$ .

In Example 5, note that all even integers are related to zero, as $(0, ± 2)$ , $(0, ± 4)$ etc., lie in $R$ and no odd integer is related to $0$ , as $(0, ± 1)$ , $(0, ± 3)$ etc., do not lie in $R$ . Similarly, all odd integers are related to one and no even integer is related to one. Therefore, the set $E$ of all even integers and the set $O$ of all odd integers are subsets of $ℤ$ satisfying following conditions:

(i) All elements of $E$ are related to each other and all elements of $O$ are related to each other.
(ii) No element of $E$ is related to any element of $O$ and vice-versa.
(iii) $E$ and $O$ are disjoint and $ℤ = E ∪ O$ .

The subset $E$ is called the equivalence class containing zero and is denoted by $[0]$ . Similarly, $O$ is the equivalence class containing $1$ and is denoted by $[1]$ . Note that $[0] ≠ [1]$ , $[0] = [2r]$ and $[1] = [2r + 1]$ , $r ∈ ℤ$ . Infact, what we have seen above is true for an arbitrary equivalence relation $R$ in a set $X$ . Given an arbitrary equivalence relation $R$ in an arbitrary set $X$ , $R$ divides $X$ into mutually disjoint subsets Ai called partitions or subdivisions of $X$ satisfying:

(i) all elements of $A_i$ are related to each other, for all i.
(ii) no element of $A_i$ is related to any element of $A_j$ , $i ≠ j$ .
(iii) $∪ A j = X$ and $A_i ∩ A_j = φ$ , $i ≠ j$ .

The subsets $A_i$ are called equivalence classes. The interesting part of the situation is that we can go reverse also. For example, consider a subdivision of the set $ℤ$ given by three mutually disjoint subsets $A_1$ , $A_2$ and $A_3$ whose union is $ℤ$ with

$A_1 = \{x ∈ ℤ : x$ is a multiple of $3\} = \{..., - 6, - 3, 0, 3, 6, ...\}$

$A_2 = \{x ∈ ℤ : x - 1$ is a multiple of $3\} = \{..., - 5, - 2, 1, 4, 7, ...\}$

$A_3 = \{x ∈ ℤ : x - 2$ is a multiple of $3\} = \{..., - 4, - 1, 2, 5, 8, ...\}$

Define a relation $R$ in $ℤ$ given by $R = \{(a, b) : 3$ divides $a - b\}$ . Following the arguments similar to those used in Example 5, we can show that $R$ is an equivalence relation. Also, $A_1$ coincides with the set of all integers in $ℤ$ which are related to zero, $A_2$ coincides with the set of all integers which are related to $1$ and $A_3$ coincides with the set of all integers in $ℤ$ which are related to $2$ . Thus, $A_1 = [0], A_2 = [1]$ and $A_3 = [2]$ . In fact, $A_1 = [3r]$ , $A_2 = [3r + 1]$ and $A_3 = [3r + 2]$ , for all $r ∈ ℤ$ .

उदाहरण 6

Let $R$ be the relation defined in the set $A = \{1, 2, 3, 4, 5, 6, 7\}$ by $R = \{(a, b) :$ both $a$ and $b$ are either odd or even $\}$ . Show that $R$ is an equivalence relation. Further, show that all the elements of the subset $\{1, 3, 5, 7\}$ are related to each other and all the elements of the subset $\{2, 4, 6\}$ are related to each other, but no element of the subset $\{1, 3, 5, 7\}$ is related to any element of the subset $\{2, 4, 6\}$ .

हल

Given any element $a$ in $A$ , both $a$ and $a$ must be either odd or even, so that $(a, a) ∈ R$ . Further, $(a, b) ∈ R ⇒$ both $a$ and $b$ must be either odd or even $⇒ (b, a) ∈ R$ . Similarly, $(a, b) ∈ R$ and $(b, c) ∈ R ⇒$ all elements $a$ , $b$ , $c$ , must be either even or odd simultaneously $⇒ (a, c) ∈ R$ . Hence, $R$ is an equivalence relation. Further, all the elements of $\{1, 3, 5, 7\}$ are related to each other, as all the elements of this subset are odd. Similarly, all the elements of the subset $\{2, 4, 6\}$ are related to each other, as all of them are even. Also, no element of the subset $\{1, 3, 5, 7\}$ can be related to any element of $\{2, 4, 6\}$ , as elements of $\{1, 3, 5, 7\}$ are odd, while elements of $\{2, 4, 6\}$ are even.

EXERCISE 1.1

1.1 Question 1

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

(i) Relation $R$ in the set $A = \{1, 2, 3, ..., 13, 14\}$ defined as $R = \{(x, y) : 3x - y = 0\}$
(ii) Relation $R$ in the set $N$ of natural numbers defined as $R = \{(x, y) : y = x + 5$ and $x < 4\}$
(iii) Relation $R$ in the set $A = \{1, 2, 3, 4, 5, 6\}$ as $R = \{(x, y) : y$ is divisible by $x\}$
(iv) Relation $R$ in the set $Z$ of all integers defined as $R = \{(x, y) : x - y$ is an integer $\}$
(v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by
- (a) $R = \{(x, y) : x$ and y work at the same place $\}$
- (b) $R = \{(x, y) : x$ and y live in the same locality $\}$
- (c) $R = \{(x, y) : x$ is exactly 7 cm taller than $y\}$
- (d) $R = \{(x, y) : x$ is wife of $y\}$
- (e) $R = \{(x, y) : x$ is father of $y\}$

1.1 Question 2

Show that the relation $R$ in the set $R$ of real numbers, defined as $R = \{(a, b) : a ≤ b2\}$ is neither reflexive nor symmetric nor transitive.

1.1 Question 3

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.

1.1 Question 4

Show that the relation $R$ in $R$ defined as $R = \{(a, b) : a ≤ b\}$ , is reflexive and transitive but not symmetric.

1.1 Question 5

Check whether the relation $R$ in $R$ defined by $R = \{(a, b) : a ≤ b3\}$ is reflexive,symmetric or transitive.

1.1 Question 6

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.

1.1 Question 7

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.

1.1 Question 8

Show that the relation $R$ in the set $A = \{1, 2, 3, 4, 5\}$ given by $R = \{(a, b) : |a - b|$ is even $\}$ , is an equivalence relation. Show that all the elements of $\{1, 3, 5\}$ are related to each other and all the elements of $\{2, 4\}$ are related to each other. But no element of $\{1, 3, 5\}$ is related to any element of $\{2, 4\}$ .

1.1 Question 9

Show that each of the relation 4 in the set $A = \{x ∈ Z : 0 ≤ x ≤ 12\}$ , given by

(i) $R = \{(a, b) : |a - b|$ is a multiple of $4\}$
(ii) $R = \{(a, b) : a = b\}$ is an equivalence relation. Find the set of all elements related to 1 in each case.

1.1 Question 10

Give an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

1.1 Question 11

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 points related to a point $P ≠ (0, 0)$ is the circle passing through $P$ with origin as centre.

1.1 Question 12

Show that the relation R defined in the set A of all triangles as $R = \{(T_1, T_2) : T_1$ is similar to $T_2\}$ , is equivalence relation. Consider three right angle triangles $T_1$ with sides $3, 4, 5$ , $T_2$ with sides $5, 12, 13$ and $T_3$ with sides $6, 8, 10$ . Which triangles among $T_1$ , $T_2$ and $T_3$ are related?

1.1 Question 13

Show that the relation $R$ defined in the set $A$ of all polygons as $R = \{(P_1, P_2) : P_1$ and $P_2$ 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 $5$ ?

1.1 Question 14

Let $L$ be the set of all lines in $XY$ plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1$ is parallel to $L_2\}$ . Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$ .

1.1 Question 15

Let $R$ be the relation in the set $\{1, 2, 3, 4\}$ given by $R = \{(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)\}$ . Choose the correct answer.

(A) $R$ is reflexive and symmetric but not transitive.
(B) $R$ is reflexive and transitive but not symmetric.
(C) $R$ is symmetric and transitive but not reflexive.
(D) $R$ is an equivalence relation.

1.1 Question 16

Let $R$ be the relation in the set $N$ given by $R = \{(a, b) : a = b - 2, b > 6\}$ . Choose the correct answer.

(A) $(2, 4) ∈ R$
(B) $(3, 8) ∈ R$
(C) $(6, 8) ∈ R$
(D) $(8, 7) ∈ R$

1.3 Types of Functions

The notion of a function along with some special functions like identity function, constant function, polynomial function, rational function, modulus function, signum function etc. along with their graphs have been given in Class XI.

Addition, subtraction, multiplication and division of two functions have also been studied. As the concept of function is of paramount importance in mathematics and among other disciplines as well, we would like to extend our study about function from where we finished earlier. In this section, we would like to study different types of functions.

Consider the functions $f_1$ , $f_2$ , $f_3$ and $f_4$ given by the following diagrams.

In Fig 1.2, we observe that the images of distinct elements of $X_1$ under the function $f_1$ are distinct, but the image of two distinct elements $1$ and $2$ of $X_1$ under $f_2$ is same, namely $b$ . Further, there are some elements like $e$ and $f$ in $X_2$ which are not images of any element of $X1$ under $f_1$ , while all elements of $X3$ are images of some elements of $X1$ under $f_3$ . The above observations lead to the following definitions:

परिभाषा 5

A function $f : X → Y$ is defined to be one-one (or injective), if the images of distinct elements of $X$ under $f$ are distinct, i.e., for every $x_1, x_2 ∈ X, f(x_1) = f(x_2)$ implies $x_1 = x_2$ . Otherwise, $f$ is called many-one.

The function $f_1$ and $f_4$ in Fig 1.2 (i) and (iv) are one-one and the function $f_2$ and $f3_3$ in Fig 1.2 (ii) and (iii) are many-one.

परिभाषा 6

A function $f : X → Y$ is said to be onto (or surjective), if every element of $Y$ is the image of some element of $X$ under f, i.e., for every $y ∈ Y$ , there exists an element $x$ in $X$ such that $f(x) = y$ .

The function $f_3$ and $f_4$ in Fig 1.2 (iii), (iv) are onto and the function $f_1$ in Fig 1.2 (i) is not onto as elements $e$ , $f$ in $X_2$ are not the image of any element in $X1$ under $f_1$ .

परिभाषा 7

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

The function $f_4$ in Fig 1.2 (iv) is one-one and onto.

उदाहरण 7

Let $A$ be the set of all $50$ students of Class X in a school. Let $f : A → ℕ$ be function defined by $f(x) =$ roll number of the student $x$ . Show that $f$ is one-one but not onto.

हल

No two different students of the class can have same roll number. Therefore, $f$ must be one-one. We can assume without any loss of generality that roll numbers of students are from $1$ to $50$ . This implies that $51$ in $ℕ$ is not roll number of any student of the class, so that $51$ can not be image of any element of $X$ under $f$ . Hence, $f$ is not onto.

उदाहरण 8

Show that the function $f : ℕ → ℕ$ , given by $f(x) = 2x$ , is one-one but not onto.

हल

The function $f$ is one-one, for $f(x_1) = f(x_2) ⇒ 2x_1 = 2x_2 ⇒ x_1 = x_2$ . Further, f is not onto, as for $1 ∈ ℕ$ , there does not exist any $x$ in $ℕ$ such that $f(x) = 2x = 1$ .

उदाहरण 9

Prove that the function $f : ℝ → ℝ$ , given by $f(x) = 2x$ , is one-one and onto.

हल

$f$ is one-one, as $f(x_1) = f(x_2) ⇒ 2x_1 = 2x_2 ⇒ x_1 = x_2$ . Also, given any real number $y$ in $ℝ$ , there exists $\frac{y}{2}$ in $ℝ$ such that $f(\frac{y}{2}) = 2⋅(\frac{y}{2}) = y$ . Hence, $f$ is onto.

उदाहरण 10

Show that the function $f : ℕ → ℕ$ , given by $f (1) = f (2) = 1$ and $f(x) = x - 1$ , for every $x > 2$ , is onto but not one-one.

हल

$f$ is not one-one, as $f(1) = f(2) = 1$ . But $f$ is onto, as given any $y ∈ ℕ, y ≠ 1$ , we can choose $x$ as $y + 1$ such that $f(y + 1) = y + 1 - 1 = y$ . Also for $1 ∈ ℕ$ , we have $f(1) = 1$ .

उदाहरण 11

Show that the function $f : ℝ → ℝ$ , defined as $f(x) = x^2$ , is neither one-one nor onto.

हल

Since $f(-1) = 1 = f (1)$ , $f$ is not one-one. Also, the element $-2$ in the co-domain $ℝ$ is not image of any element $x$ in the domain $ℝ$ (Why?). Therefore $f$ is not onto.

उदाहरण 12

Show that $f : ℕ → ℕ$ , given by

f(x) = \begin{cases} x+1,& \text{if }x\text{ is odd},\\ x-1,& \text{if }x\text{ is even}, \end{cases}

is both one-one and onto.

हल

Suppose $f(x_1) = f(x_2)$ . Note that if $x_1$ is odd and $x_2$ is even, then we will have $x_1 + 1 = x_2 - 1$ , i.e., $x_2 - x_1 = 2$ which is impossible. Similarly, the possibility of $x_1$ being even and $x_2$ being odd can also be ruled out, using the similar argument. Therefore, both $x_1$ and $x_2$ must be either odd or even. Suppose both $x_1$ and $x_2$ are odd. Then $f(x_1) = f(x_2) ⇒ x_1 + 1 = x_2 + 1 ⇒ x_1 = x_2$ . Similarly, if both $x_1$ and $x_2$ are even, then also $f(x_1) = f(x_2) ⇒ x_1 - 1 = x_2 - 1 ⇒ x_1 = x_2$ . Thus, $f$ is one-one. Also, any odd number $2r + 1$ in the co-domain $ℕ$ is the image of $2r+ 2$ in the domain $ℕ$ and any even number $2r$ in the co-domain $ℕ$ is the image of $2r - 1$ in the domain $ℕ$ . Thus, $f$ is onto.

उदाहरण 13

Show that an onto function $f : \{1, 2, 3\} → \{1, 2, 3\}$ is always one-one.

हल

Suppose $f$ is not one-one. Then there exists two elements, say $1$ and $2$ in the domain whose image in the co-domain is same. Also, the image of $3$ under $f$ can be only one element. Therefore, the range set can have at the most two elements of the co-domain $\{1, 2, 3\}$ , showing that $f$ is not onto, a contradiction. Hence, $f$ must be one-one.

उदाहरण 14

Show that a one-one function $f : \{1, 2, 3\} → \{1, 2, 3\}$ must be onto.

हल

Since $f$ is one-one, three elements of $\{1, 2, 3\}$ must be taken to 3 different elements of the co-domain $\{1, 2, 3\}$ under $f$ . Hence, $f$ has to be onto.

EXERCISE 1.2

1.2 Question 1

Show that the function $f : ℝ_* → ℝ_*$ defined by $f(x) = \frac{1}{x}$ is one-one and onto, where $ℝ_*$ is the set of all non-zero real numbers. Is the result true, if the domain $ℝ_*$ is replaced by $ℕ$ with co-domain being same as $ℝ_*$ ?

1.2 Question 2

Check the injectivity and surjectivity of the following functions:

(i) $f : ℕ → ℕ$ given by $f(x) = x^2$
(ii) $f : ℤ → ℤ$ given by $f(x) = x^2$
(iii) $f : ℝ → ℝ$ given by $f(x) = x^2$
(iv) $f : ℕ → ℕ$ given by $f(x) = x^3$
(v) $f : ℤ → ℤ$ given by $f(x) = x^3$

1.2 Question 3

Prove that the Greatest Integer Function $f : ℝ → ℝ$ , given by $f(x) = [x]$ , is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to $x$ .

1.2 Question 4

Show that the Modulus Function $f : ℝ → ℝ$ , given by $f(x) = |x|$ , is neither one- one nor onto, where $|x|$ is $x$ , if $x$ is positive or $0$ and $|x|$ is $-x$ , if $x$ is negative.

1.2 Question 5

Show that the Signum Function $f : ℝ → ℝ$ , given by is neither one-one nor onto.

1.2 Question 6

Let $A = \{1, 2, 3\}$ , $B = \{4, 5, 6, 7\}$ and let $f = \{(1, 4), (2, 5), (3, 6)\}$ be a function from $A$ to $B$ . Show that $f$ is one-one.

1.2 Question 7

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) $f : ℝ → ℝ$ defined by $f(x) = 3 - 4x$
(ii) $f : ℝ → ℝ$ defined by $f(x) = 1 + x^2$

1.2 Question 8

Let $A$ and $B$ be sets. Show that $f : A \times B → B \times A$ such that $f(a, b) = (b, a)$ is bijective function.

1.2 Question 9

Let $f : ℕ → ℕ$ be defined by

f(n) = \begin{cases} \frac{n+1}{2}, & \text{if } n \text{ is odd}\\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} \text{for all } n ∈ ℕ.

State whether the function $f$ is bijective. Justify your answer.

1.2 Question 10

Let $A = ℝ - \{3\}$ and $B = ℝ - \{1\}$ . Consider the function $f : A → B$ defined by $f(x) = \frac{x-2}{x-3}$ . Is $f$ one-one and onto? Justify your answer.

1.2 Question 11

Let $f : ℝ → ℝ$ be defined as $f(x) = x^4$ . Choose the correct answer.

(A) $f$ is one-one onto
(B) $f$ is many-one onto
(C) $f$ is one-one but not onto
(D) $f$ is neither one-one nor onto.

1.2 Question 12

Let $f : ℝ → ℝ$ be defined as $f(x) = 3x$ . Choose the correct answer.

(A) $f$ is one-one onto
(B) $f$ is many-one onto
(C) $f$ is one-one but not onto
(D) $f$ is neither one-one nor onto.

1.4 Composition of Functions and Invertible Function

परिभाषा 8

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 $gof(x) = g(f(x))$ , $∀ x ∈ A$ .

उदाहरण 15

Let

$f : \{2, 3, 4, 5\} → \{3, 4, 5, 9\}$ and

$g : \{3, 4, 5, 9\} → \{7, 11, 15\}$

be functions defined as

$f(2) = 3$ , $f(3) = 4$ , $f(4) = f(5) = 5$ and

$g(3) = g(4) = 7$ and $g(5) = g(9) = 11$ .

Find $gof$ .

हल

We have

$gof(2) = g(f(2)) = g(3) = 7$ ,

$gof(3) = g(f(3)) = g(4) = 7$ ,

$gof(4) = g(f(4)) = g(5) = 11$

and $gof(5) = g (5) = 11$ .

उदाहरण 16

Find $gof$ and $fog$ , if $f : ℝ → ℝ$ and $g : ℝ → ℝ$ are given by $f (x) = cos x$ and $g (x) = 3x^2$ . Show that $gof ≠ fog$ .

हल

We have $gof (x) = g (f(x)) = g(cos(x)) = 3 (cos(x))^2 = 3 cos^2(x)$ . Similarly, $fog(x) = f(g(x)) = f(3x^2) = cos(3x^2)$ . Note that $3cos^2(x) ≠ cos(3x^2)$ , for $x = 0$ . Hence, $gof ≠ fog$ .

परिभाषा 9

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 $f^{-1}$ .

Thus, 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. This fact significantly helps for proving a function $f$ to be invertible by showing that $f$ is one-one and onto, specially when the actual inverse of $f$ is not to be determined.

उदाहरण 17

Let $f : ℕ → Y$ be a function defined as $f(x) = 4x + 3$ , where, $Y = \{y ∈ ℕ: y = 4x + 3$ for some $x ∈ ℕ\}$ . Show that $f$ is invertible. Find the inverse.

हल

Consider an arbitrary element $y$ of $Y$ . By the definition of $Y$ , $y = 4x + 3$ , for some $x$ in the domain ℕ.

This shows that $x = \frac{(y − 3)}{4}$ . Define $g : Y → N$ by $g (y) = \frac{(y − 3)}{4}$ .

Now, $gof(x) = g (f(x)) = g(4x + 3) = \frac{(4x + 3 − 3)}{4} = x$ and

$fog(y) = f(g(y)) = f(\frac{(y-3)}{4}) = \frac{4(y-3)}{4} + 3 = y - 3 + 3 = y$ .

This shows that $gof = I_ℕ$ and $fog = I_Y$ , which implies that $f$ is invertible and $g$ is the inverse of $f$ .

Miscellaneous Examples

उदाहरण 18

If $R_1$ and $R_2$ are equivalence relations in a set $A$ , show that $R_1 ∩ R_2$ is also an equivalence relation.

हल

Since $R_1$ and $R_2$ are equivalence relations, $(a, a) ∈ R_1$ , and $(a, a) ∈ R_2$ $∀ a ∈ A$ .

This implies that $(a, a) ∈ R_1 ∩ R_2, ∀ a$ , showing $R_1 ∩ R_2$ is reflexive.

Further, $(a, b) ∈ R_1 ∩ R_2$

$⇒ (a, b) ∈ R_1$ and $(a, b) ∈ R_2$

$⇒ (b, a) ∈ R_1$ and $(b, a) ∈ R_2$

$⇒ (b, a) ∈ R_1 ∩ R_2$ , hence, $R_1 ∩ R_2$ is symmetric.

Similarly, $(a, b) ∈ R_1 ∩ R_2$ and $(b, c) ∈ R_1 ∩ R_2$

$⇒ (a, c) ∈ R_1$ and $(a, c) ∈ R_2$

$⇒ (a,c) ∈ R_1 ∩ R_2$ .

This shows that $R_1 ∩ R_2$ is transitive.

Thus, $R_1 ∩ R_2$ is an equivalence relation.

उदाहरण 19

Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $xv = yu$ . Show that $R$ is an equivalence relation.

हल

Clearly, $(x, y) R (x, y), ∀ (x, y) ∈ A$ , since $xy = yx$ .

This shows that $R$ is reflexive.

Further, $(x, y) R (u, v)$

$⇒ xv = yu ⇒ uy = vx$ and hence $(u, v) R (x, y)$ .

This shows that $R$ is symmetric.

Similarly, $(x, y) R (u, v)$ and $(u, v) R (a,b)$

$⇒ xv = yu$ and $ub = va$

$⇒ xv\frac{a}{u} = yu\frac{a}{u}$

$⇒ xv\frac{b}{v} = yu\frac{a}{u}$

$⇒ xb = ya$ and hence $(x, y) R (a, b)$ .

Thus, $R$ is transitive.

Thus, $R$ is an equivalence relation.

उदाहरण 20

Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ .

Let $R_1$ be a relation in $X$ given by

$R_1 = \{(x, y) : x - y$ is divisible by $3\}$

and $R_2$ be another relation on $X$ given by

$R_2 = \{(x, y): \{x, y\} ⊂ \{1, 4, 7\}\}$ or $\{x, y\} ⊂ \{2, 5, 8\} or \{x, y\} ⊂ \{3, 6, 9\}\}$ .

Show that $R_1 = R_2$ .

हल

Note that the characteristic of sets $\{1, 4, 7\}$ , $\{2, 5, 8\}$ and $\{3, 6, 9\}$ is that difference between any two elements of these sets is a multiple of $3$ .

Therefore, $(x, y) ∈ R_1$

$⇒ x - y$ is a multiple of $3$

$⇒ \{x, y\} ⊂ \{1, 4, 7\}$ or $\{x, y\} ⊂ \{2, 5, 8\}$ or $\{x, y\} ⊂ \{3, 6, 9\}$

$⇒ (x, y) ∈ R_2$ . Hence, $R_1 ⊂ R_2$ .

Similarly, $\{x, y\} ∈ R_2$

$⇒ \{x, y\} ⊂ \{1, 4, 7\}$ or $\{x, y\} ⊂ \{2, 5, 8\}$ or $\{x, y\} ⊂ \{3, 6, 9\}$

$⇒ x - y$ is divisible by $3$

$⇒ \{x, y\} ∈ R_1$ . This shows that $R_2 ⊂ R_1$ .

Hence, $R_1 = R_2$ .

उदाहरण 21

Let $f : X → Y$ be a function. Define a relation $R$ in $X$ given by $R = \{(a, b): f(a) = f(b)\}$ . Examine whether $R$ is an equivalence relation or not.

हल

For every $a ∈ X, (a, a) ∈ R$ , since $f (a) = f (a)$ , showing that $R$ is reflexive.

Similarly, $(a, b) ∈ R$

$⇒ f (a) = f (b)$

$⇒ f (b) = f (a)$

$⇒ (b, a) ∈ R$ . Therefore, $R$ is symmetric.

Further, $(a, b) ∈ R$ and $(b, c) ∈ R$

$⇒ f (a) = f (b)$ and $f (b) = f (c)$

$⇒ f (a) = f (c)$

$⇒ (a, c) ∈ R$ , which implies that $R$ is transitive.

Hence, $R$ is an equivalence relation.

उदाहरण 22

Find the number of all one-one functions from set $A = \{1, 2, 3\}$ to itself.

हल

One-one function from $\{1, 2, 3\}$ to itself is simply a permutation on three symbols $1, 2, 3$ . Therefore, total number of one-one maps from $\{1, 2, 3\}$ to itself is same as total number of permutations on three symbols $1, 2, 3$ which is $3! = 6$ .

उदाहरण 23

Let $A = \{1, 2, 3\}$ . Then show that the number of relations containing $(1, 2)$ and $(2, 3)$ which are reflexive and transitive but not symmetric is three.

हल

The smallest relation R1 containing $(1, 2)$ and $(2, 3)$ which is reflexive and transitive but not symmetric is $\{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$ . Now, if we add the pair $(2, 1)$ to $R_1$ to get $R_2$ , then the relation $R_2$ will be reflexive, transitive but not symmetric. Similarly, we can obtain $R_3$ by adding $(3, 2)$ to $R_1$ to get the desired relation. However, we can not add two pairs $(2, 1), (3, 2)$ or single pair $(3, 1)$ to $R_1$ at a time, as by doing so, we will be forced to add the remaining pair in order to maintain transitivity and in the process, the relation will become symmetric also which is not required. Thus, the total number of desired relations is three.

उदाहरण 24

Show that the number of equivalence relation in the set $\{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is two.

हल

The smallest equivalence relation $R_1$ containing $(1, 2)$ and $(2, 1)$ is $\{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$ . Now we are left with only 4 pairs namely $(2, 3)$ , $(3, 2)$ , $(1, 3)$ and $(3, 1)$ . If we add any one, say $(2, 3)$ to $R_1$ , then for symmetry we must add $(3, 2)$ also and now for transitivity we are forced to add $(1, 3)$ and $(3, 1)$ . Thus, the only equivalence relation bigger than $R_1$ is the universal relation. This shows that the total number of equivalence relations containing $(1, 2)$ and $(2, 1)$ is two.

उदाहरण 25

Consider the identity function $I_ℕ : ℕ → ℕ$ defined as $I_ℕ(x) = x ∀ x ∈ ℕ$ . Show that although $I_ℕ$ is onto but $I ℕ + I ℕ : ℕ → ℕ$ defined as

$(I_ℕ + I_ℕ)(x) = I_ℕ(x) + I_ℕ(x) = x + x = 2x$ is not onto.

हल

Clearly $I_ℕ$ is onto. But $I_ℕ + I_ℕ$ is not onto, as we can find an element $3$ in the co-domain $ℕ$ such that there does not exist any $x$ in the domain $ℕ$ with $(I_ℕ + I_ℕ)(x) = 2x = 3$ .

उदाहरण 26

Consider a function $f : [0,\frac{π}{2}] → ℝ$ given by $f(x) = sin(x)$ and $g : [0,\frac{π}{2}] → ℝ$ given by $g(x) = cos(x)$ . Show that $f$ and $g$ are one-one, but $f + g$ is not one-one.

हल

Since for any two distinct elements $x_1$ and $x_2$ in $[0,\frac{π}{2}]$ , $sin(x_1) ≠ sin(x_2)$ and $cos(x_1) ≠ cos(x_2)$ , both $f$ and g must be one-one.

But $(f + g)(0) = sin(0) + cos(0) = 1$ and $(f + g)(\frac{π}{2}) = sin(\frac{π}{2}) + cos(\frac{π}{2}) = 1$ .

Therefore, $f + g$ is not one-one.

Miscellaneous Exercise on Chapter 1

1.M Question 1

Show that the function $f : ℝ → \{x ∈ ℝ : - 1 < x < 1\}$ defined by $f(x) = \frac{x}{1+|x|}$ , $x ∈ R$ is one one and onto function.

1.M Question 2

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

1.M Question 3

Given a non empty set $X$ , consider $P(X)$ which is the set of all subsets of $X$ . Define the relation $R$ in $P(X)$ as follows:

For subsets $A$ , $B$ in $P(X)$ , $ARB$ if and only if $A ⊂ B$ . Is $R$ an equivalence relation on $P(X)$ ? Justify your answer.

1.M Question 4

Find the number of all onto functions from the set $\{1, 2, 3,......, n\}$ to itself.

1.M Question 5

Let $A = \{- 1, 0, 1, 2\}$ , $B = \{- 4, - 2, 0, 2\}$ and $f, g : A → B$ be functions defined by $f(x) = x^2 - x, x ∈ A$ and $g(x) = 2|x-\frac{1}{2}|-1, x ∈ A$ . Are $f$ and $g$ equal? Justify your answer.

(Hint: One may note that two functions $f : A → B$ and $g : A → B$ such that $f(a) = g(a) ∀ a ∈ A$ , are called equal functions).

1.M Question 6

Let $A = \{1, 2, 3\}$ . Then number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is

(A) 1
(B) 2
(C) 3
(D) 4

1.M Question 7

Let $A = \{1, 2, 3\}$ . Then number of equivalence relations containing $(1, 2)$ is

(A) 1
(B) 2
(C) 3
(D) 4

Summary

In this chapter, we studied different types of relations and equivalence relation, composition of functions, invertible functions and binary operations. The main features of this chapter are as follows:

Empty relation is the relation $R$ in $X$ given by $R = φ ⊂ X \times X$ .
Universal relation is the relation $R$ in $X$ given by $R = X \times X$ .
Reflexive relation $R$ in $X$ is a relation with $(a, a) ∈ R ∀ a ∈ X$ .
Symmetric relation $R$ in $X$ is a relation satisfying $(a, b) ∈ R$ implies $(b, a) ∈ R$ .
Transitive relation $R$ in $X$ is a relation satisfying $(a, b) ∈ R$ and $(b, c) ∈ R$ implies that $(a, c) ∈ R$ .
Equivalence relation $R$ in $X$ is a relation which is reflexive, symmetric and transitive.
Equivalence class [a] containing $a ∈ X$ for an equivalence relation $R$ in $X$ is the subset of $X$ containing all elements $b$ related to $a$ .
A function $f : X → Y$ is one-one (or injective) if $f(x_1) = f(x_2) ⇒ x_1 = x_2 ∀ x_1, x_2 ∈ X$ .
A function $f : X → Y$ is onto (or surjective) if given any $y ∈ Y$ , $∃ x ∈ X$ such that $f(x) = y$ .
A function $f : X → Y$ is one-one and onto (or bijective), if $f$ is both one-one and onto.
Given a finite set $X$ , a function $f : X → X$ is one-one (respectively onto) if and only if $f$ is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set

ऐतिहासिक पृष्ठभूमि

फलन की संकल्पना, R. Descartes (सन् 1596-1650 ई.) से प्रारंभ हो कर एक लंबे अंतराल में विकसित हुई है। Descartes ने सन् 1637 ई. में अपनी पांडुलिपि “Geometrie” में शब्द ‘फलन’ का प्रयोग, ज्यामितीय वक्रों, जैसे अतिपरवलय (Hyperbola), परिवलय (Parabola) तथा दीर्घवृत्त (Ellipse), का अध्ययन करते समय, एक चर राशि $x$ के धन पूर्णांक घात $x^n$ के अर्थ में किया था। James Gregory (सन् 1636-1675 ई.) ने अपनी वृफ़ति “Vera Circuli et Hyperbolae Quadratura” (सन् 1667 ई.) में, फलन को एक ऐसी राशि माना था, जो किसी अन्य राशि पर बीजीय अथवा अन्य संक्रियाओं को उत्तरोत्तर प्रयोग करने से प्राप्त होती है। बाद में G. W. Leibnitz (1646-1716 ई.) नें 1673 ई. में लिखित अपनी पांडुलिपि “Methodus tangentium inversa, seu de functionibus” में शब्द ‘फलन’ को किसी ऐसी राशि के अर्थ में प्रयोग किया, जो किसी वक्र के एक बिंदु से दूसरे बिंदु तक इस प्रकार परिवर्तित होती रहती है, जैसे वक्र पर बिंदु के निर्देशांक, वक्र की प्रवणता, वक्र की स्पर्शी तथा अभिलंब परिवर्तित होते हैं। तथापि अपनी वृफ़ति “Historia” (1714 ई.) में Leibnitz ने फलन को एक चर पर आधारित राशि के रूप में प्रयोग किया था। वाक्यांश ’ $x$ का फलन’ प्रयोग में लाने वाले वे सर्वप्रथम व्यक्ति थे। John Bernoulli (1667-1748 ई.) ने सर्वप्रथम 1718 ई. में संकेतन (Notation) $φx$ को वाक्यांश ’ $x$ का फलन’ को प्रकट करने के लिए किया था। परंतु फलन को निरूपित करने के लिए प्रतीकों, जैसे $f$ , $F$ , $φ$ , $ψ$ … का व्यापक प्रयोग Leonhard Euler (1707-1783 ई.) द्वारा 1734 ई. में अपनी पांडुलिपि “Analysis Infinitorium”. के प्रथम खण्ड में किया गया था। बाद में Joeph Louis Lagrange (1736-1813 ई.) ने 1793 ई. में अपनी पांडुलिपि “Theorie des functions analytiques” प्रकाशित की, जिसमें उन्होंने विश्लेषणात्मक (Analytic) फलन के बारे में परिचर्चा की थी तथा संकेतन $f(x)$ , $F(x)$ , $φ(x)$ आदि का प्रयोग $x$ के भिन्न-भिन्न फलनों के लिए किया था। तदोपरांत Lejeunne Dirichlet (1805-1859 ई.) ने फलन की परिभाषा दी। जिसका प्रयोग उस समय तक होता रहा जब तक वर्तमान काल में फलन की समुच्चय सैद्धांतिक परिभाषा का प्रचलन नहीं हुआ, जो Georg Cantor (1845-1918 ई) द्वारा विकसित समुच्चय सिद्धांत के बाद हुआ। वर्तमान काल में प्रचलित फलन की समुच्चय सैद्धांतिक परिभाषा Dirichlet द्वारा प्रदत्त फलन की परिभाषा का मात्र अमूर्तीकरण (Abstraction) है।