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1. Relations and Functions

3D GEOMETRY NOTES

Suresh142
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1. Relations and Functions

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  1. CHAPTER -1 RELATIONS AND FUNCTIONS Blue Print (As Per PU Board) Three Marks Qns MCQs Five Mark Qns Total Marks NCERT EXERCISE NCERT EXERCISE Relations Examples 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. 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. 3.Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive. 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. 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 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}. Exercise 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} 1 Mathematics

  2. (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 ≤ b2 } 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 ≤ b3} is reflexive symmetric or transitive. 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. 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 numbers of pages} is an equivalence relation. 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 elements of {2, 4}. 9.Show that each of the relation R 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. 10.Given 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. 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. 2 Mathematics

  3. 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. 12.Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related? 13.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? 14.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. 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. 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 Functions Functions Example 1.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. 2.Show that the function f: N  N, given by f(x) = 2x, is one-one but not onto. 3.Prove that the function f: R  R, given by f(x) = 2x, is one-one and onto. 4.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. 5.Show that the function f: R  R, defined as f(x) = x2, is neither one-one nor onto.     x 1, x 1,  if x is odd if x is even 6.Show that f: N  N, given by is both one-one and onto, f(x) 3 Mathematics

  4. Exercise 1 x   1.Show that the function f: defined by is one-one and onto, where R is the set of all R R f(x) * * * non-zero real numbers. Is the result true, if the domain R is replaced by N with co-domain being same * as R ? * 2.Check the injectivity and surjectivity of the following functions: (i) f: N  N given by f(x) = x2 (ii) f: Z  Z given by f(x) = x2 (iii) f: R  R given by f(x) = x2 (iv) f: N  N given by f(x) = x3 (v) f: Z  Z given by f(x) = x3 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. 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,if x 0,if x 1,if x  0 0 0 5.Show that the Signum Function f: R  R, given by    f x Is neither one-one nor onto. 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. 7.In each of the following cases, state whether the function in 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 + x2 8.Let A and B be sets. Show that f: A  B  B  A such that f(a, b) = (b, a) is bijective function. n 1,if nisodd 2 f n n 2        9.Let f: N  N be defined by     foralln N. State whether the function f is ,if niseven bijective. Justify your answer.       x x 3  2 10.Let A = R – {3} and B = R – {1}. Consider the function f: A  B defined by  . Is f one- f(x) one and onto? Justify your answer. 11.Let f: R  R be defined as f(x) = x4. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto 4 Mathematics

  5. (C) f is one-one but not onto (D) f is neither one-one nor onto 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 Example 1.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. 2.Find gof and fog, if f: R  R and g: R  R are given by f(x) = cos x and g(x) = 3x2. Show that gof  fog.                         ; IA (x) = x,  x  A,       is defined 3x 4 5x 7  7 5 3 5 3 5 7 5 7 5        3.Show that if is defined by and f :R R g:R R f(x)         7x 4 5x 3 3 5      by , then fog = IA and gof = IB, where, A R ,B R g(x) IB (x) = x,  x  B are called identity functions on sets A and B, respectively. 4.Show that if f: A  B and g: B  C are one-one, then gof: A  C is also one-one. 5.Show that if f: A  B and g: B  C are onto, then gof: A  C is also onto. 6.Consider functions f and g such that composite gof is defined and is one-one. Are f and g both necessarily one-one. 7.Are f and g both necessarily onto, if gof is onto? 8.Let f: {1, 2, 3}  {a, b, c} be one-one and onto function given by f(1) = a, f(2) = b and f(3) = c. Show that there exists a function g: {a, b, c}  {1, 2, 3} such that gof = IX and fog = IY, where, X = {1, 2, 3} and Y = {a, b, c}. 9.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. 10.Let Y = {n2 : n  N}  N. Consider f: N  Y as f(n) = n2. Show that f is invertible. Find the inverse of f. 11.Let f : N  R be a function defined as f (x) = 4x2 + 12x + 15. Show that f: N  S, where, S is the range of f, is invertible. Find the inverse of f. 12.Consider f : N  N, g: N  N and h: N  R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z,  x, y and z in N. Show that ho(gof) = (hog) of. 5 Mathematics

  6. 13.Consider f: {1, 2, 3}  {a, b, c} and g: {a, b, c}  {apple, ball, cat} defined as f(1) = a, f(2) = b, f(3) = c, g(a) = apple, g(b) = ball g(c) = cat. Show that f, g and gof are invertible. Find out f–1, g–1 and (gof)–1 and show that (gof)–1 = f–10 g–1. 14.Let S = {1, 2, 3}. Determine whether the functions f: S  S defined as below have inverse. Find f–1, if it exists. (a) f = {(1, 1), (2, 2), (3, 3)} (b) f = {(1, 2), (2, 1), (3, 1)} (c) f = {(1, 3), (3, 2), (2, 1)} Exercise 1.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. 2.Let f, g and h be functions from R to R. Show that (f + g) 0 h = foh + goh (f . g) 0 h = (foh) . (goh) 3.Find gof and fog, if (ii) f(x) = 8x3 and g(x) = x1/3. 2 x 3 (i) f(x) = | x | and g(x) = | 5x – 2|   6x 4     4x 3 2 3    4.If show that fof(x) = x, for all . What is the inverse of f? f(x) ,x , 5.State with reason whether following functions have inverse (i) f: {1, 2, 3 4}  {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)} (ii) g: {5, 6, 7, 8}  {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)} (iii) h:{2, 3, 4, 5}  {7, 9, 11, 13} with h= {(2, 7), (3, 9), (4, 11), (5, 13)} x  6.Show that f: [–1, 1]  R, given by  is one-one. Find the inverse of the function f: [–1, 1] f(x)   x 2  Range f. 7.Consider f: R  R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f. 8.Consider f : R+ [4, ) given by f(x) =x2 + 4. Show that f is invertible with the inverse f–1 of f given by f–1(y) = y 4  , where R+ is the set of all non-negative real numbers. 9.Consider f: R+ [–5, ) given by f(x) = 9x2 + 6x –5. Show that f is invertible with  1 f y 3           y 6 1     . 10.Let f: X  Y be an invertible function. Show that f has unique inverse. 6 Mathematics

  7. 11.Consider f: {1, 2, 3}  {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f–1 and show that (f–1)–1 = f. 12.Let f: X  Y be an invertible function. Show that the inverse of f–1 is f, i.e., (f–1)–1 = f. Multiple Choice Questions Multiple Choice Questions 1.A relation R in a set A, If each element of A is related to every element of A”, then R is called (A) empty relation (B) universal relation (C) Trivial relations (D) none of these. 2.Both the empty relation and the universal relation is (A) empty relation (B) universal relation (C) Trivial relations. (D) equivalence relations. 3.Let A be the set of all students of a boys school. Then the relation R in A given by R = {(a, b) : a is sister of b} is (A) empty relation (B) transitive relation (C) symmetric relations (D) reflexive relations. 4.A relation R in the set A is called a reflexive relations, if (A) (a, a)  R, for every a  A, (B) (a, a)  R, at least one a  A (C) if (a, b)  R implies that (b, a)  R, for all a, b  A (D) if (a, b)and (b, c)  R implies that (a, c)  R, for all a, b, c  A 5.A relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1,3)}. Then R is (A) reflexive and symmetric (B) reflexive and transitive (C) reflexive , symmetric and transitive (D) reflexive but neither symmetric nor transitive 6.A relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is (A) reflexive and symmetric (B) symmetric but not transitive (C) symmetric and transitive (D) neither symmetric nor transitive. 7.A relation R in the set {1, 2, 3} given that R = {(1, 2), (2, 1), (1, 1)} is (A) transitive but not symmetric (B) symmetric but not transitive (C) symmetric and transitive (D) neither symmetric nor transitive. 7 Mathematics

  8. 8.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. 9.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, 6)  R. 10.Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is (A) symmetric but not transitive (B) transitive but not symmetric (C) neither symmetric nor transitive (D) both symmetric and transitive. 11.If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)},then R is (A) reflexive (B) transitive (C) symmetric (D) none of these. 12.Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and only if l is perpendicular to m l, m  L. Then R is (A) reflexive (B) symmetric (C) transitive (D) none of these. 13.Let R be the relation in the set {1, 2, 3, 4} given by R = {(2, 2),(1, 1),(4, 4),(3, 3)}. 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. 14.Let W denote the words in the English dictionary. Define the relation R by R={(x, y)  W × W : the words x and y have at least one letter in common}. Then R is (A) not reflexive, symmetric and transitive (B) reflexive, symmetric and not transitive (C) reflexive, symmetric and transitive (D) reflexive, not symmetric and transitive. 15.Let S = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is (A) 1 (B) 2 (C) 3 (D) 4 16.The number of equivalence relation in the set {1, 2, 3} containing (1, 2) and (2, 1) is (A) 5 (B) 2 (C) 4 (D) 3 8 Mathematics

  9. 17. Let S = {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 18.If a relation R on the set {1, 2, 3} be defined by R = {(1, 1)},then R is (A) symmetric but not transitive (B) transitive but not symmetric (C) symmetric and transitive (D) neither symmetric nor transitive. 19.Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is (A) Reflexive and symmetric (B) Transitive and symmetric (C) Equivalence (D) Reflexive, transitive but not symmetric 20.Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b  a, b  T. Then R is (A) reflexive but not transitive (B) transitive but not symmetric (C) equivalence (D) none of these 21.Let A = {2, 3, 4, 5} and B = {36, 45, 49, 60, 77, 90} and let R be the relation ‘ is factor of’ from A to B Then the range of R is the set (A) {60} (B) {36, 45, 60, 90} (C) {49, 77} (D) {49, 60, 77} 22.The maximum number of equivalence relation on the set are A = {1, 2, 3} are (A) 1 (B) 2 (C) 3 (D) 5 23.Let us define a relation R in R as aRb if a  b. Then R is (A) an equivalence relation (B) reflexive, transitive but not symmetric (C) symmetric, transitive but not reflexive (D) neither transitive nor reflexive but symmetric . 24.A relation R in set A = {1, 2, 3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the following ordered pair in R shall be removed to make it an equivalence relation in A? (A) (1, 1) (B) (1, 2) (C) (2, 2) (D) (3, 3) 25.Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (1, 2), (2, 3), (3, 3)}. Then R is (A) reflexive but not symmetric (B) reflexive but not transitive (C) symmetric and transitive (D) neither symmetric, nor transitive. 26.If a relation R on the set {1, 2, 3} be defined by R = {(1, 1),(2, 2)},then R is (A) symmetric but not transitive (B) transitive but not symmetric (C) symmetric and transitive (D) neither symmetric nor transitive. 9 Mathematics

  10. 27.Let f : R  R be defined by f(x) = x4, x  R. Then (A) f is one-one but not onto (B) f is one-one and onto (C) f is many-one onto (D) f is neither one-one nor onto . 28.Let f : R  R be defined by f(x) = 3x, x  R. Then (A) f is one-one but not onto (B) f is one-one and onto (C) f is many-one onto (D) f is neither one-one nor onto . 29.Let f : R  R be defined by f(x) = x3, x  R. Then (A) f is one-one but not onto (B) f is one-one and onto (C) f is many-one onto 30.Let f : R  R be defined by f(x) = 1 (D) f is neither one-one nor onto . x , x  R. Then f is (A) one-one (B) onto (C) bijective (D) f is not defined. 31.Let f : R  R defined by f(x) = 2x + 6 which is a bijective mapping then f−1(x) is given by (A) x 2– 3 (B) 2x + 6 (C) x – 3 (D) 6x + 2 32.If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is (A) 720 (B) 120 (C) 0 (D) None of these 33.Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is (A) nP2 (B) 2n – 2 (C) 2n  1 (D) None of these 34.If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one mappings from A to B is (A) 720 (B) 120 (C) 0 (D) none of these 35.A contains 5 elements and the set B contains 6 elements, then the number of onto mappings from A to B is (A) 720 (B) 120 (C) 0 (D) none of these 36.Let N be the set of natural numbers and the function f : N  N be defined by f(n) = 2n + 3  n  N. Then f is (A) surjective (B) Injective (C) bijective (D) none of these 37.Which of the following functions from Z into Z are bijections? (A) f(x) = x3 (D) f(x) = x2 + 1 (B) f(x) = x + 2 (C) f(x) = 2x + 1 38.Let f : R  R be defined by f(x) = 3x − 4. Is invertible. Then f –1(x) is given by 10 Mathematics

  11. (A) x 4 (B) x 3 (C) 3x + 4 (D) None of these 4 3 39.Let S = {a, b, c} and T = {1, 2, 3} then which of the following functions f from S to T, f –1 exists. (A) f = {(a, 3), (b, 2), (c, 1)} (B) f = {(a, 1), (b, 1), (c, 1)} (C) f = {(a, 2), (b, 1), (c, 1)} (D) f = {(a, 1), (b, 2), (c, 1)} 40.Find the number of all one-one functions from set A = {1, 2, 3, 4} to itself. (A) 8 (B) 24 (C) 16 (D) 256 Statement Prob Statement Problems lems 41.Statement 1: A relation R = {(1, 1), (1, 2), (2, 1)} defined on the set A = {1, 2, 3} is transitive. Statement 2: A relation R on the set A is transitive if (a, b) and (b, c)  R, then(a, c)  R (A) Statement 1 is true and Statement 2 is false. (B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for Statement 1 (C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is false and Statement 2 is true. 42.Let f : {1, 2, 3 }  {1, 2, 3} is a function, Statement 1: If f is one-one, then f must be onto. Statement 2: If f is onto, then f must be one-one. Choose the correct answer. (A) Statement 1 is true, and Statement 2 is false. (B) Statement 1 is false, and Statement 2 is true. (C) Statement 1 is true, and Statement 2 is true (D) Statement 1 is false, and Statement 2 is false. 43.Statement 1: Let f : R  R be defined by f(x) = 3x is bijective. Statement 2: A function f : A  B is a bijective function if f is one-one and onto (A) Statement 1 is true and Statement 2 is false. (B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for Statement 1 (C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is false and Statement 2 is false. 44.Statement 1: Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R is an identity relation. Statement 2: R is an equivalence relation, then R is reflexive ,symmetric and transitive. (A) Statement 1 is true and Statement 2 is false. (B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for Statement 1 (C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 11 Mathematics

  12. (D) Statement 1 is false and Statement 2 is false. 45.Assertion (A): In set A = {1, 2, 3} a relation R defined as R = {(1, 1), (2, 2)} is reflexive. Reason (R): A relation R is reflexive in set A if (a, a)  R for all a  A (A) A is false but R is true (B) A is true and R is true (C) A is true but R is false (D) A is false and R is false. 46.Assertion (A): In set A = {1, 2, 3} a relation R in set A , given as R = {(1, 2)} is transitive. Reason (R): A singleton relation is transitive (A) A is false but R is true (B) A is true and R is false (C) A is true but R is false (D) A is true and R is true. 47.Assertion(A): If n(A) = 3, then the number of reflexive relations on A is 3 Reason (R): A relation R on the set A is reflexive if (a, a)  R,  a  A (A) A is false but R is true (B) A is true and R is false (C) A is true but R is true (D) A is false and R is false 48.Assertion(A): A relations R = {(a, a), (b, b), (b, c), (c, c)} defined on the set A = {a, b, c} is symmetric Reason(R): A relation R on the set A is symmetric if (a, b)  R  (b, a)  R (A) A is true but R is true (B) A is false and R is true (C) A is true but R is false (D) A is false and R is false 0, if x is rational 1, if x is irrational   49.Statement I: The function f : R  R defined as    is bijective f x Statement 2: A function is said to be bijective if it is both one one and onto (A) Statement 1 is true, and Statement 2 is false (B) Statement 1 is false, and Statement 2 is true (C) Statement 1 is true, and Statement 2 is true (D) Statement 1 is false, and Statement 2 is false 50.Statement 1: A function f : A  B, can not be an onto function if n(A) < n(B). Statement 2: A function f is onto if every element of co-domain has at least one pre-image in the domain (A) Statement 1 is true and Statement 2 is false. (B) Statement 1 is true and Statement 2 is true, Statement 2 is correct explanation for Statement 1 (C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is false and Statement 2 is false. 51.Consider the set A containing 3 elements. Then, the total number of injective functions from A onto itself is _________ 12 Mathematics

  13. 52.Set A has 3 elements, and set B has 4 elements. Then the number of injective mappings that can be defined from A to B is_________ 53.Let A = {1, 2, 3} and B = {a, b}.Then the number of surjections from A into B is_______ 54.The number of equivalence relations containing (2, 1) on the set A = {1, 2, 3} is_______ 55.A contains 4 elements and the set B contains 5 elements, then the number of onto mappings from A to B is ______ 56.If f : {2, 8}  {2, 2, 4}, for the following figure f is (A) f is one-one but not onto (B) f is one-one and onto (C) f is neither one-one nor onto (D) f is not a function 57.If f : R → R ,then graph of the function is (A) f is one-one but not onto (B) f is one-one and onto (C) f is neither one-one nor onto . (D) f is onto but not one-one 58.If f : R  R, then graph of the function is (A) f is one-one but not onto (B) f is one-one and onto (C) f is neither one-one nor onto (D) f is onto but not one-one 59.The maximum number of equivalence relations on the set A = {1, 2} is ------ 13 Mathematics

  14. 60.Given set A ={1, 2, 3} and a relation R = {(3, 1), (1, 3),(3,3)}, the relation R will be (A) reflexive if (1, 1) is added (B) symmetric if (2, 3) is added (C) transitive if (1, 1) is added (D) symmetric if (3, 2) is added. 61.Let X = {–1, 0, 1}, Y = {0, 2} and a function f : X  Y defined by y = 2x4, is (A) one-one and onto (B) one-one into (C) many-one onto (D) many-one into. 62.Let A be the set of all 100 students of Class XII in a college. Let f : A  N be function defined by f (x) = roll number of the student Class XII. (A) f is neither one-one nor onto (B) f is one-one but not onto (C) f is not one-one but onto (D) none of these. 63.Statement 1:If R and S are two equivalence relations on a set A, then R ∩ S is also an Equivalence relation on A. Statement 2: The union of two equivalence relations on a set is not necessarily relation on the set. Statement 3: The inverse of an equivalence relation is an equivalence relation. (A) All 3 Statements are true (B) 1 and 2 Statements are true but 3 false (C) All 3 Statements are false (D) 1 and 3 Statements are true but 2 false. 64.The number of bijective functions from set A to itself is 120,then A contains ______ elements . 65.Let R be an equivalence relation on a finite set A having n elements. Then ,the number of ordered pair in R is (A) < n (B) > or = n (C) < or = n (D) none of these Two Mark Questions Two Mark Questions 1.Define a reflexive relation and give an example of it. 2.Define a symmetric relation and give an example of it. 3.Define a transitive relation and give an example of it. 4.Define an equivalence relation and give an example of it. 5.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. 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. 14 Mathematics

  15. 7.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. 8.Show that an onto function f : {1, 2, 3}  {1, 2, 3} is always one-one. 9.Show that one-one function f : {1, 2, 3}  {1, 2, 3} is always onto. 10.Let A = {1, 2, 3}, B = {4, 5, 6, 7} and f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one. 11.Show that the Signum Function f : R  R, is neither one-one nor onto. 12.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. 13.If f : R  R is defined by f(x) = 3x – 2. Show that f is one-one. 14.If f : N  N given by f(x) = x2 check whether f is one-one. Justify your answer. 15.If f : Z  Z given by f(x) = x3 check whether f is one-one and onto. 16.If f : R  R given by f(x) = x3 check whether f is one-one and onto. 17.If f : Z  Z given by f(x) = x3 check whether f is one-one and onto. 18.Show that the function f : N  N, given by f(x) = 2x is one-one but not onto. 19.Show that the function given by f(1) = f(2) = 1 and f(x) = x – 1, for every x > 2, is onto but not one-one. 20.If f : R  R given by f(x) = 3x check whether f is one-one and onto 21.Prove f : R  R that given by f(x) = x3 is onto. 22.Let : {2, 3, 4, 5}  {3, 4, 5, 9} and g : {3, 4, 5, 9}  {7, 11, 15} be functions defined 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. 23.Let f : {1, 3, 4}  {1, 2, 5} and g : {1, 2, 5}  {1, 3} given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)} write down gof. 24.If f : A  B and g : B  C are one-one then show that gof : A  C is also one-one. 25.If f : A  B and g : B  C are onto then show that gof : A  C is also onto. 26.State with reason whether f : {1, 2, 3, 4}  {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)} has inverse. 27.State with reason whether f : {5, 6, 7, 8}  {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)} has inverse. 28.State with reason whether h : {2, 3, 4, 5}  {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)} has inverse. Three Mark Questions Three Mark Questions 15 Mathematics

  16. 1.A relation R on the set A = {1, 2, 3......14} is defined as R = {(x, y) : 3x – y =0}. Determine whether R is reflexive, symmetric and transitive. 2.A relation R in the set N of natural number defined as R = {(x, y) : y = x + 5 and x < 4}. Determine whether R is reflexive, symmetric and transitive. 3.A relation ‘R’ is defined on the set A = {1, 2, 3, 4, 5} as R = {(x, y) : y is divisible by x}. Determine whether R is reflexive, symmetric, 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. 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. 6.Relation R in the set Z of all integers is defined as R = {(x, y) : x – y is an integer}. Determine whether R is reflexive, symmetric and transitive. 7.Determine whether R, in the set A of human beings in a town at a particular time is given by R = {(x, y) : x and y work at the same place} 8.Show that the relation R in R, the set of reals defined as R = {(a, b) : a  b} is reflexive and transitive but not symmetric. 9.Show that the relation R on the set of real numbers R is defined by R = {(a, b) : a  b2} is neither reflexive nor symmetric nor transitive. 10.Check whether the relation R in R the set of real numbers defined as R = {(a, b) : a  b3} is reflexive, symmetric and transitive. 11.Show the relation R in the set Z of integers given by R = {(a, b) : 2 divides (a – b)} is an equivalence relation. 12.Show the relation R in the set Z of integers given by R = {(a, b) : (a – b) is divisible by 2} is an equivalence relation. 13.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. 14.Show that the relation R on the set A of point on coordinate plane given by R = {(P, Q) distance OP = OQ, where O is origin is an equivalence relation. 15.Show that the relation R on the set A = {x  Z : 0  x  12} given by R = {(a, b)/ |a  b| : is a multiple of 4} is an equivalence relation. 16 Mathematics

  17. 16.Show that the relation R on the set A = {x  Z : 0  x  12} given by R = {(a, b) : a = b} is an equivalence relation. 17.Let T be the set of triangles with R  a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Show that R is an equivalence relation. 18.Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive. 19.Let L be the set of all lines in the XY plane and R is the relation on L b 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. 20.Show that the relation R defined in the set A of polygons as R = {(P1, P2) : P1 and P2 have same number of side} is an equivalence relation. 21.If R1 and R2 are two equivalence relations on a set, is R1 R2 also an equivalence relation? Justify your answer. 22.If R1 and R2 are two equivalence relations on a set, then prove that R1 R2 is also an equivalence relation. 23.Find gof and fog if f : R  R and g : R  R are given by f(x) = cos x and g(x) = 3x2. Show that gof  fog. 24.If f and g are functions from R  R defined by f(x) = sin x and g(x) = x2. Show that gof  fog. Five Five Mark Questions Mark Questions 1.Consider f : R  R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f. 2.Consider f : R  R given by f(x) = 10x + 7. Show that f is invertible. Find the inverse of f. 3.If f : R  R given by f(x) = x2 check whether f is one-one and onto. Justify your answer. 4.Show that the modulus function f : R  R given by f(x) = |x| is neither one-one nor onto 5.Prove that the greatest integer function f : R  R given by f(x) = |x| is neither one-one nor onto.     x, x 0,  x 0 6.Show that the Signum function f : R  R defined by    is neither one-one nor onto. f x  x 0 Justify your answer.       n 1 2 n 2 if n is odd 7.Let f : N  N defined by    f n . State whether f is bijective. Justify your answer if n is even 17 Mathematics

  18. 1 x 8.Show that the function f : R0 R0, given by    is one-one and onto, where R0 is the set of all f x non-zero real numbers. Is the result true, if the domain R0 is replaced by N with co-domain being same as R0? 9.Let A and B be sets. Show that f : A  B  B  A such that f(a, b) = (b, a) is bijective function       x x 3  2 10.Consider the function f : A  B defined by    . Is f one-one and onto? f x 11.If f : R  R is defined by f(x) = 1 + x2, then show that f is neither one-one nor onto 12.Consider f : R+ [4, ) given by f(x) = x2 + 4. Show that f is invertible. Find the inverse of f. 4 f :R R 3   4 f :R 3       4x  be a function defined by define       13.Let . Find the inverse of the function f x 3x 4        Range of f. 18 Mathematics

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