Page 1 vThere 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. — G. H. HARDY v 1.1 Introduction Recall that the notion of relations and functions, domain, co-domain and range have been introduced in Class XI along with different types of specific real valued functions and their graphs. The concept of the term ‘relation’ in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school. Then some of the examples of relations from A to B are (i) {(a, b) ? A × B: a is brother of b}, (ii) {(a, b) ? A × B: a is sister of b}, (iii) {(a, b) ? A × B: age of a is greater than age of b}, (iv) {(a, b) ? A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination}, (v) {(a, b) ? A × B: a lives in the same locality as b}. However, abstracting from this, we define mathematically a relation R from A to B as an arbitrary subset of A × B. If (a, b) ? R, we say that a is related to b under the relation R and we write as a R b. In general, (a, b) ? R, we do not bother whether there is a recognisable connection or link between a and b. As seen in Class XI, functions are special kind of relations. In this chapter, we will study different types of relations and functions, composition of functions, invertible functions and binary operations. Chapter 1 RELATIONS AND FUNCTIONS Lejeune Dirichlet (1805-1859) 2019-20 Page 2 vThere 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. — G. H. HARDY v 1.1 Introduction Recall that the notion of relations and functions, domain, co-domain and range have been introduced in Class XI along with different types of specific real valued functions and their graphs. The concept of the term ‘relation’ in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school. Then some of the examples of relations from A to B are (i) {(a, b) ? A × B: a is brother of b}, (ii) {(a, b) ? A × B: a is sister of b}, (iii) {(a, b) ? A × B: age of a is greater than age of b}, (iv) {(a, b) ? A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination}, (v) {(a, b) ? A × B: a lives in the same locality as b}. However, abstracting from this, we define mathematically a relation R from A to B as an arbitrary subset of A × B. If (a, b) ? R, we say that a is related to b under the relation R and we write as a R b. In general, (a, b) ? R, we do not bother whether there is a recognisable connection or link between a and b. As seen in Class XI, functions are special kind of relations. In this chapter, we will study different types of relations and functions, composition of functions, invertible functions and binary operations. Chapter 1 RELATIONS AND FUNCTIONS Lejeune Dirichlet (1805-1859) 2019-20 MATHEMATICS 2 1.2 Types of Relations In this section, we would like to study different types of relations. We know that a relation in a set A is a subset of A × A. Thus, the empty set f and A × A are two extreme relations. For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by R = {(a, b): a – b = 10}. This is the empty set, as no pair (a, b) satisfies the condition a – b = 10. Similarly, R' = {(a, b) : | a – b | = 0} is the whole set A × A, as all pairs (a, b) in A × A satisfy | a – b | = 0. These two extreme examples lead us to the following definitions. Definition 1 A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = f ? A × A. Definition 2 A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A. Both the empty relation and the universal relation are some times called trivial relations. Example 1 Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R' = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation. Solution Since the school is boys school, no student of the school can be sister of any student of the school. Hence, R = f, 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 3 meters. This shows that R' = A × A is the universal relation. Remark In Class XI, we have seen two ways of representing a relation, namely raster method and set builder method. However, a relation R in the set {1, 2, 3, 4} defined by R = {(a, b) : b = a + 1} is also expressed as a R b if and only if b = a + 1 by many authors. We may also use this notation, as and when convenient. If (a, b) ? R, we say that a is related to b and we denote it as a R b. One of the most important relation, which plays a significant role in Mathematics, is an equivalence relation. To study equivalence relation, we first consider three types of relations, namely reflexive, symmetric and transitive. Definition 3 A relation R in a set A is called (i) reflexive, if (a, a) ? R, for every a ? A, (ii) symmetric, if (a 1 , a 2 ) ? R implies that (a 2 , a 1 ) ? R, for all a 1 , a 2 ? A. (iii) 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. 2019-20 Page 3 vThere 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. — G. H. HARDY v 1.1 Introduction Recall that the notion of relations and functions, domain, co-domain and range have been introduced in Class XI along with different types of specific real valued functions and their graphs. The concept of the term ‘relation’ in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school. Then some of the examples of relations from A to B are (i) {(a, b) ? A × B: a is brother of b}, (ii) {(a, b) ? A × B: a is sister of b}, (iii) {(a, b) ? A × B: age of a is greater than age of b}, (iv) {(a, b) ? A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination}, (v) {(a, b) ? A × B: a lives in the same locality as b}. However, abstracting from this, we define mathematically a relation R from A to B as an arbitrary subset of A × B. If (a, b) ? R, we say that a is related to b under the relation R and we write as a R b. In general, (a, b) ? R, we do not bother whether there is a recognisable connection or link between a and b. As seen in Class XI, functions are special kind of relations. In this chapter, we will study different types of relations and functions, composition of functions, invertible functions and binary operations. Chapter 1 RELATIONS AND FUNCTIONS Lejeune Dirichlet (1805-1859) 2019-20 MATHEMATICS 2 1.2 Types of Relations In this section, we would like to study different types of relations. We know that a relation in a set A is a subset of A × A. Thus, the empty set f and A × A are two extreme relations. For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by R = {(a, b): a – b = 10}. This is the empty set, as no pair (a, b) satisfies the condition a – b = 10. Similarly, R' = {(a, b) : | a – b | = 0} is the whole set A × A, as all pairs (a, b) in A × A satisfy | a – b | = 0. These two extreme examples lead us to the following definitions. Definition 1 A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = f ? A × A. Definition 2 A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A. Both the empty relation and the universal relation are some times called trivial relations. Example 1 Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R' = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation. Solution Since the school is boys school, no student of the school can be sister of any student of the school. Hence, R = f, 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 3 meters. This shows that R' = A × A is the universal relation. Remark In Class XI, we have seen two ways of representing a relation, namely raster method and set builder method. However, a relation R in the set {1, 2, 3, 4} defined by R = {(a, b) : b = a + 1} is also expressed as a R b if and only if b = a + 1 by many authors. We may also use this notation, as and when convenient. If (a, b) ? R, we say that a is related to b and we denote it as a R b. One of the most important relation, which plays a significant role in Mathematics, is an equivalence relation. To study equivalence relation, we first consider three types of relations, namely reflexive, symmetric and transitive. Definition 3 A relation R in a set A is called (i) reflexive, if (a, a) ? R, for every a ? A, (ii) symmetric, if (a 1 , a 2 ) ? R implies that (a 2 , a 1 ) ? R, for all a 1 , a 2 ? A. (iii) 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. 2019-20 RELATIONS AND FUNCTIONS 3 Definition 4 A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive. Example 2 Let T be the set of all triangles in a plane with R a relation in T given by R = {(T 1 , T 2 ) : T 1 is congruent to T 2 }. Show that R is an equivalence relation. Solution R is reflexive, since every triangle is congruent to itself. Further, (T 1 , T 2 ) ? R ? T 1 is congruent to T 2 ? T 2 is congruent to T 1 ? (T 2 , T 1 ) ? R. Hence, R is symmetric. Moreover, (T 1 , T 2 ), (T 2 , T 3 ) ? R ? T 1 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. Example 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. Solution 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 ? (L 2 , L 1 ) ? 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. Example 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. Solution 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. Example 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. Solution 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. Fig 1.1 2019-20 Page 4 vThere 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. — G. H. HARDY v 1.1 Introduction Recall that the notion of relations and functions, domain, co-domain and range have been introduced in Class XI along with different types of specific real valued functions and their graphs. The concept of the term ‘relation’ in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school. Then some of the examples of relations from A to B are (i) {(a, b) ? A × B: a is brother of b}, (ii) {(a, b) ? A × B: a is sister of b}, (iii) {(a, b) ? A × B: age of a is greater than age of b}, (iv) {(a, b) ? A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination}, (v) {(a, b) ? A × B: a lives in the same locality as b}. However, abstracting from this, we define mathematically a relation R from A to B as an arbitrary subset of A × B. If (a, b) ? R, we say that a is related to b under the relation R and we write as a R b. In general, (a, b) ? R, we do not bother whether there is a recognisable connection or link between a and b. As seen in Class XI, functions are special kind of relations. In this chapter, we will study different types of relations and functions, composition of functions, invertible functions and binary operations. Chapter 1 RELATIONS AND FUNCTIONS Lejeune Dirichlet (1805-1859) 2019-20 MATHEMATICS 2 1.2 Types of Relations In this section, we would like to study different types of relations. We know that a relation in a set A is a subset of A × A. Thus, the empty set f and A × A are two extreme relations. For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by R = {(a, b): a – b = 10}. This is the empty set, as no pair (a, b) satisfies the condition a – b = 10. Similarly, R' = {(a, b) : | a – b | = 0} is the whole set A × A, as all pairs (a, b) in A × A satisfy | a – b | = 0. These two extreme examples lead us to the following definitions. Definition 1 A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = f ? A × A. Definition 2 A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A. Both the empty relation and the universal relation are some times called trivial relations. Example 1 Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R' = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation. Solution Since the school is boys school, no student of the school can be sister of any student of the school. Hence, R = f, 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 3 meters. This shows that R' = A × A is the universal relation. Remark In Class XI, we have seen two ways of representing a relation, namely raster method and set builder method. However, a relation R in the set {1, 2, 3, 4} defined by R = {(a, b) : b = a + 1} is also expressed as a R b if and only if b = a + 1 by many authors. We may also use this notation, as and when convenient. If (a, b) ? R, we say that a is related to b and we denote it as a R b. One of the most important relation, which plays a significant role in Mathematics, is an equivalence relation. To study equivalence relation, we first consider three types of relations, namely reflexive, symmetric and transitive. Definition 3 A relation R in a set A is called (i) reflexive, if (a, a) ? R, for every a ? A, (ii) symmetric, if (a 1 , a 2 ) ? R implies that (a 2 , a 1 ) ? R, for all a 1 , a 2 ? A. (iii) 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. 2019-20 RELATIONS AND FUNCTIONS 3 Definition 4 A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive. Example 2 Let T be the set of all triangles in a plane with R a relation in T given by R = {(T 1 , T 2 ) : T 1 is congruent to T 2 }. Show that R is an equivalence relation. Solution R is reflexive, since every triangle is congruent to itself. Further, (T 1 , T 2 ) ? R ? T 1 is congruent to T 2 ? T 2 is congruent to T 1 ? (T 2 , T 1 ) ? R. Hence, R is symmetric. Moreover, (T 1 , T 2 ), (T 2 , T 3 ) ? R ? T 1 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. Example 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. Solution 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 ? (L 2 , L 1 ) ? 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. Example 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. Solution 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. Example 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. Solution 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. Fig 1.1 2019-20 MATHEMATICS 4 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 A i 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 n A j = f, 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. Example 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}. 2019-20 Page 5 vThere 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. — G. H. HARDY v 1.1 Introduction Recall that the notion of relations and functions, domain, co-domain and range have been introduced in Class XI along with different types of specific real valued functions and their graphs. The concept of the term ‘relation’ in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let A be the set of students of Class XII of a school and B be the set of students of Class XI of the same school. Then some of the examples of relations from A to B are (i) {(a, b) ? A × B: a is brother of b}, (ii) {(a, b) ? A × B: a is sister of b}, (iii) {(a, b) ? A × B: age of a is greater than age of b}, (iv) {(a, b) ? A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination}, (v) {(a, b) ? A × B: a lives in the same locality as b}. However, abstracting from this, we define mathematically a relation R from A to B as an arbitrary subset of A × B. If (a, b) ? R, we say that a is related to b under the relation R and we write as a R b. In general, (a, b) ? R, we do not bother whether there is a recognisable connection or link between a and b. As seen in Class XI, functions are special kind of relations. In this chapter, we will study different types of relations and functions, composition of functions, invertible functions and binary operations. Chapter 1 RELATIONS AND FUNCTIONS Lejeune Dirichlet (1805-1859) 2019-20 MATHEMATICS 2 1.2 Types of Relations In this section, we would like to study different types of relations. We know that a relation in a set A is a subset of A × A. Thus, the empty set f and A × A are two extreme relations. For illustration, consider a relation R in the set A = {1, 2, 3, 4} given by R = {(a, b): a – b = 10}. This is the empty set, as no pair (a, b) satisfies the condition a – b = 10. Similarly, R' = {(a, b) : | a – b | = 0} is the whole set A × A, as all pairs (a, b) in A × A satisfy | a – b | = 0. These two extreme examples lead us to the following definitions. Definition 1 A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = f ? A × A. Definition 2 A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A. Both the empty relation and the universal relation are some times called trivial relations. Example 1 Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R' = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation. Solution Since the school is boys school, no student of the school can be sister of any student of the school. Hence, R = f, 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 3 meters. This shows that R' = A × A is the universal relation. Remark In Class XI, we have seen two ways of representing a relation, namely raster method and set builder method. However, a relation R in the set {1, 2, 3, 4} defined by R = {(a, b) : b = a + 1} is also expressed as a R b if and only if b = a + 1 by many authors. We may also use this notation, as and when convenient. If (a, b) ? R, we say that a is related to b and we denote it as a R b. One of the most important relation, which plays a significant role in Mathematics, is an equivalence relation. To study equivalence relation, we first consider three types of relations, namely reflexive, symmetric and transitive. Definition 3 A relation R in a set A is called (i) reflexive, if (a, a) ? R, for every a ? A, (ii) symmetric, if (a 1 , a 2 ) ? R implies that (a 2 , a 1 ) ? R, for all a 1 , a 2 ? A. (iii) 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. 2019-20 RELATIONS AND FUNCTIONS 3 Definition 4 A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive. Example 2 Let T be the set of all triangles in a plane with R a relation in T given by R = {(T 1 , T 2 ) : T 1 is congruent to T 2 }. Show that R is an equivalence relation. Solution R is reflexive, since every triangle is congruent to itself. Further, (T 1 , T 2 ) ? R ? T 1 is congruent to T 2 ? T 2 is congruent to T 1 ? (T 2 , T 1 ) ? R. Hence, R is symmetric. Moreover, (T 1 , T 2 ), (T 2 , T 3 ) ? R ? T 1 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. Example 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. Solution 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 ? (L 2 , L 1 ) ? 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. Example 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. Solution 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. Example 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. Solution 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. Fig 1.1 2019-20 MATHEMATICS 4 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 A i 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 n A j = f, 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. Example 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}. 2019-20 RELATIONS AND FUNCTIONS 5 Solution 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. 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} 2. 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. 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. 4. Show that the relation R in R defined as R = {(a, b) : a = b}, is reflexive and transitive but not symmetric. 5. Check whether the relation R in R defined by R = {(a, b) : a = b 3 } is reflexive, symmetric or transitive. 2019-20Read More

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