BADHAN ACADEMY OF MATHS CONTACT FOR HOMETUTOR, FOR 11 th 12 th B.B.A, B.C.A, DIPLOMA,SAT,CAT SSC 1.2 RELATIONS CHAPTER AT GLANCE

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1 BADHAN ACADEMY OF MATHS CONTACT FOR HOMETUTOR, FOR th 2 th B.B.A, B.C.A, DIPLOMA,SAT,CAT SSC.2 RELATIONS CHAPTER AT GLANCE. RELATIONS : Let A and B be two non- empty sets. A relation from set A to set B is a subset of A B. Thus, if R is a relations from A to B then R A B. Also, if ( a, b ) R, then we say that a is R related to be and denote this by a R b. In particular, if B = A then the subsets of A A are called relations from the set A to the set or simply as relations in the set A Illustrations: (i) Let A = {,3,5,7}, B = { 6,8 }, Let R be the relations is less than from A to B. R 6, R8, 3R6, 3R8, 5R6, 5R8, 7R8. Equivalently, R = { (,6), (,8), (3,6), (3,8), (5,6),(5,8), ( 7, 8)}. (ii) Let A = {,2,3,..,32}. Let R be the relation is one fourth of ;; in A. R4, 2R8,3R2,4R6, 5R20, 6R24, 7R28, 8R32 Equivalent, R = { (,4), (2,8), (3,2), (4,6), (5,20), (6,24), (7,28), (8,32) }, 2. DOMAIN AND RANGE : If R is a relation from A to B, then the set of first elements of elements in R is called the domain of R and the set of second element of elements in R is called the range of R. Symbolically, Domain of R = { x : ( x, y ) R } ; Range of R { y: (x, y } R}. Domain of R is a subset of A and range of R is a subset of B. For example, R = { 4, 7), ( 5,8), (6, 0 }. A = {,2,3.4,5,6} to the set B = { 6,7,8,9, 0 }. Domain of R = { 4,5,6}. this is a subset of A.Range of R = { 7,8, 0}. This is a subset of B. 3. NUMBER OF RELATIONS: Let A and B be two non- empty finite sets consisting of m and n elements respectively. A B contain mn ordered pairs Total number of subsets of A B is 2 mn Since each relation from A to B is a subset of A B, the total number of relations from A to B is 2 mn 4, TYPES OF RELATIONS: (i) A relation R in a set A is called the universal relation in a if R = A A. for example, if A = { 2, 6 }, then the universal relation in A is the set { (2,2), (2,6), (6,2), (6,6)}. (ii) A relation R is a set A is called the identity relation in A if R= { (a,a) ; a A }.For example if A = ( a,b,c) } then the identity relation in A is the set { (a,a), (b,b), (c, c)}. (iii) a relation R in a set A is called a void relation in A if R = O/. For example if A = {,2,3} and let R be the relation defined by a R b if a b = 2. The relation R = O/ A A is a void relation. (iv) A relation R in a set called a reflexive relation if ( a, a) R a A, i.e.., a Ra a A. For example, if A = { 2, 4, 7 } then the relation {(2,2), (4,4), (7,7)} is reflexive. (v) A relation R in a set A is called a symmetric relation if (a,b) R, i.e.., a Rb implies b Ra. For example, if A = { 2,4,7}, then the relation {(2,4), (4,2),(7,7)} is symmetric.

2 BADHAN ACADEMY OF MATHS CONTACT FOR HOMETUTOR, FOR th 2 th B.B.A, B.C.A, DIPLOMA,SAT,CAT SSC (vi) A relation R in a set A is called transitive if (a,b) (b,c) R then ( a, c) R, i.e., a Rb, brc Implies a Rc. For example, if A = { 2,4,7}, then the relation { 2,4),(4,7),(2,7), (4,4)} is transitive. (vii) A relation R in a set A is called an equivalence relation if R is reflexive, symmetric and transitive. For an equivalence relation R in A, we have (i) ara a A (ii)a Rb bra (iii)arb and b Rc arc For example, if A = {,3,4,7}, then the relation { (,),(,3),(3,),(3,3),(4,4),(7,7),(4,7),(7,4) is an equivalence relation (vii) A relation R in a set A is called an anti- symmetric relation if (a,b), (b,a) R, then a = b. (ix) If R is a relation from A to B, then the inverse relation of R is a relation from b to A and is defined as { (y,x): (x, y ) R }, The inverse relation in A,, then R = {(4,), (5,9), (2,0)}is the inverse relation of R. (x) Let R and S be two relation from the sets A to B to C respectively, we define a relation So R from A to C as follows : ( a,c) So R if (a,b) R and ( b, c) S for some b B. This relation SoR is called the composition of the relations R and S. For example, let A = {2,3,4}, B= {,5,7}, C = {,3,6,8}. Let R = { (2,), (2,5), (3,5), (4,), (4,7) } be a relation from A to B. Let S = { (,3), (5,3), (5,6), (7,6), (7,8)} be a relation from B to C. SoR = {(2,3),(5,3), (3,3), (4,3),(4,6), (4,8)} is a relation from A to C Note: ( 4,6),(4,8) SoR because (4,7) R and (7,6),(7,8) S. 5. CONGRUEN MODULOM m : Let m be an arbitrary but fixed integer. Two integers a and b are said to be congruence modulo m if a b is divisible by m. We write this as a b mod m. For example 24 3 mod 7 because 24-3= 2 is divisible by 7. Example : Show that congruence modulo m is an equivalence relation in the set Z. Solution : Let R denotes the relation congruence module m. Let a Z a a = 0 is divisible by m. a a mod m i.e., a Ra R is reflexive Let ( a, b ) R. a b mod m m/a b, m / - ( b a ), m /b a b a mod m, ( b,a) R R is symmetric Let ( a, b), ( b, c) R a b mod m, b c mod m m/ ( a b, m / b c ) m/ ( a b ) ( b c), m /a c a c mof m, ( a, c ) R R is transitive R is an equivalence relation 6. EQUIVALENCE CLASS: Let R be an equivalence in a set A. If a A, then the subset {x A:xRa} of A is called the equivalence class corresponding to a and it is denoted by [ a] Properties : (i) a [a ] a A (ii) a [b] if and only if [a] = [b]

3 BADHAN ACADEMY OF MATHS CONTACT FOR HOMETUTOR, FOR th 2 th B.B.A, B.C.A, DIPLOMA,SAT,CAT SSC (iii) If a,b A, then either [a] [b] = O/ or [a] = [b] Example : Let A = {,2,3} and R = { (,), (2,2), (3,3), (,2), (2,),(2,3),(3,2), (3,), (,3)} is an equivalence Relation in A. [] = {,2,3} because (,) (2,),(3,) are in R [2] = { 2,,3} because (2,2), (,2), (3,2) are in R [3] = {3, 2,} because (3,3), (2,3), (,3) are in R Example 2 : Let A = {,2,3 } and R = { (,), (2,2), (3,3), (,2), (2,) } is an equivalence relation in A [] = {,2} because (,) (2,) are in R [2] = { 2,} because ( 2,2), (,2) are in R [3] = { 3} because (3,3} is in R SOME SPECIAL TIPS ( ) ( ). Total number of relation from set A to set B is equal to 2 nanb. 2. The universal relation on a non-empty set is always reflexive, symmetric and transitive. 3. The identity relation on a non-empty set is always reflexive, symmetric and transitive 4. The identity relation on a non-empty set is always anti symmetric. 5. If R is a relation from A to B and S is a relation from B to C then ( ) ROS = S OR. 6. For two relations R and S, the composite relation RoS, may be void relation. 7. A relation R in the set A is symmetric if and only if R = R. OBJECTIVE QUESTION SET -. Let A = {,2,3, 45} and R be the relation is square of in A. which of the following is false? (i) R = {(,), (4,2), (9,3), (6,4), (25,5), (36,6)} (ii) Domain of R = {(,4,9,6,25,36)} (iii) Range of R = {(,2,3,4,5,6)} (iv) At least one is false 2. In number of element in sets A and B are m and n respectively, then number of relations from A to be is () 2 m+ n (2) 2 mn (3) m + n (4) mn 3. Let R be a reflexive relation in a finite set having n element and let there be m ordered pairs in R. Then () m n (2)m n (3) m = n (4) none of these 4. If R is a relation from a set p to set Q, then () R P Q (2) R Q P (3) R = P Q (4) R =P Q 5. If A is a finite set having n elements, then number of relations which can be defined is A is () 2 n 2 2 (2) n (3) 2 n n (4) n 6. If the relation R : A R, where A = {,2,3} and B = {,3,5} is defined by A = {(x,y):x<y, x

4 BADHAN ACADEMY OF MATHS CONTACT FOR HOMETUTOR, FOR th 2 th B.B.A, B.C.A, DIPLOMA,SAT,CAT SSC A, y B }, then () R = { (,3), (,5), ( 2,5), (3,5) } (2) R = {(,),(,5),(2,3),(3,5) } (3) R = { (3,), (5,),(3,2),(5,3)} (4) R = { (,), (5,), (3,2), (5,3)} 7. If n/m means that n is a factor m, the relation I in Z {0} is () reflexive and symmetric (2) symmetric and transitive (3) reflexive, symmetric and transitive (4) reflexive, transitive and not symmetric 8. Let R {(x,y) } : x + y = x,y R} be a relation in R. The relation R is () reflexive (2) symmetric (3) transitive (4) anti-symmetric 9. Let X be a family of sets and R be a relation in X defined by A is disjoint from B, The relation R is () reflexive (2) symmetric (3) transitive (4) anti-symmetric 0. For real numbers x and y, we define x R y if x y + 5 is an irrational number. The relation R is () reflexive (2) symmetric (3) transitive (4) none of these. Let denotes the set of all straight lines in a plane. Let a relation R be defined by Rβ α β αβ, L. The relation r is () reflexive (2) symmetric (3) transitive (4) none of these 2. Let R be a relation in the set N of natural numbers defined by the relation n Rm n is a factor of m The relation R is () reflexive and symmetric only (2) symmetric and transitive only (3) reflexive and transitive only (4) an equivalence relation 3. Two point A and B in a plane are related if OA= OB, where O is a fixed Point. This relation is () reflexive but not symmetric (2) symmetric but not transitive (3) an equivalence relation (4) none of these 4. Let A be the set of first ten natural numbers and let R be a relation in A defined by ( x, y ) R if an Only if x + 2y = 0. Which of the following is false? () R = { (2,4), (4,3), (6,2), (8,)} (2) domain of R = {(2,4,6,8) } (3) range of R = { (,2,3,4)} (4) at least one is false 5. Let A={,2,3,4} and let R = {(2,2), (3,3), (4,4), (,2)} be a relation in A. Then R is () reflexive (2) symmetric (3) transitive (4) none of these 6. A relation R is defined in the set Z of integers as follows : ( x, y ) R if x + y = 9. which of the Following is false? () R = { (0,3), (0,-3), (3,0), (-3, 0)} (2) domain of R = { -3,0,3} (3) range of R = {-3,0,3} (4) at least one is false 7. An integer m is said to be related to another integer n if m is a multiple o n. This relation is () reflexive and symmetric (2) reflexive an transitive (3) symmetric and transitive (4) an equivalence relation a b 5 8. The relation R defined in A = {,2,3} by a Rb if which of the following is false? () R = { (,), (2,2), (3,3), (2,), (,2), (2,3), (3, 2) }

5 BADHAN ACADEMY OF MATHS CONTACT FOR HOMETUTOR, FOR th 2 th B.B.A, B.C.A, DIPLOMA,SAT,CAT SSC (2) R = R (3) domain of R = {,2,3} (4) range of R = {5} 2 9. Let R be a relation in N defined by R = { ( + x, + x ) : x 5, x N}. Which of the following is false? () R = { (2,2), (3,5), (4,0), (5,7), (6,25)} (2) domain of R = { 2,3,4,5,6} (3) range of R = { 2,5,0,7,26} (4) at least one is false 20. Let A = {,2,3 }, B = {, 3,5}. If relation R from A to B is given by { (,3), (2,5), (3,3) }, then is () { ( 3,3), (3, ), (5, 3 )} (2) { (,3), (2,5), (3, 3)} (3) { (3,3), (4,3), (5,4),(3,4)} (4) none of these 2. The relation R defined in the set A = {,2,3,4,5} by R { x,y) : x y < 6 } is given by () { {,), (2,), (3,), (4,), (2, 3 )} (2) { (2,2), (3,2), (4,2), (2, 4 ) } (3) { (3,3), (4,3), ( 5,4), (3,4) } (4) none of these 22. R is a relation from {,2,3 } to { 8,0,2} defined by y = x 3. The relation () { (,8), (3, 0)} (2) {(8, ), ( 0, 3)} (3) { (8, ), (9,2), (0,3)} (4) none of these 23. Let R be a relation in N defined by R= { (x, y) : x + 2y = 8 }.The range of R is ( ) { 2,4,6} (2) {,2,3} (3) {,2,3,4,6} (4) none of these 24. IF R = { ( x, y ) : x, y Z, x + y 4 } is a relation in Z, domain of R is () { 0,,2} (2) { -2-,0} (3) { -2-,0,,2} (4) none of these 25. If A {,2,3}, then the relation r {(2,3)} in a is () symmetric and transitive only (2) symmetric only (3) transitive only (4) none of these 26. Let X be a family of sets and R be a relation in X, defined by A is disjoint from B.Then R is () reflexive (2) symmetric (3) anti-symmetric (4) transitive 27. A relation R from C to R is defined by x Ry if x = y. Which of the following is correct? () ( 2+ 3i) R 3 (2) 3 R (-3) (3) (+i) R 2 (4) ir 28. The void relation in a set A is () reflexive (2) reflexive and symmetric (3) reflexive and transitive (4) symmetric and transitive 29. If R is a relation from a set A to the set B and S is a relation from B to C, then the relation SoR () is from C to A (2) is from A to C (3) does not exit (4) none of these 30. If R be an anti-symmetric relation in a set A such that ( a, b), ( b, a) R, then () a b (2) a b (3) a = b (4) none of these 3. Let S be a non-empty set. In P (S), Let R be a relation defined as ARB A B O/. The relation R is () reflexive (2) symmetric (3) transitive (4) none of these R is R