* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Relation and Functions
Survey
Document related concepts
Functional decomposition wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Big O notation wikipedia , lookup
Abuse of notation wikipedia , lookup
Elementary mathematics wikipedia , lookup
Continuous function wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
List of first-order theories wikipedia , lookup
Dirac delta function wikipedia , lookup
Function (mathematics) wikipedia , lookup
Function of several real variables wikipedia , lookup
History of the function concept wikipedia , lookup
Transcript
1 Chapter 1 Relation and Functions 1 Mark Questions Q1. A relation R in a Set A is called ........, if each element of A is related to every element of A. Q2. Let π΄ = {π₯: β1 β€ π₯ β€ 1} and S be the subset of AΧ defined by S = {(x,y) : x2 + y2 = 1}. Is it a function? Q3. A mapping π: π β π ππ πππππππ ππ¦ π(π₯) = 2π₯ βπ₯ π π , then is f a bijection? Q4. Let X = {1, 2, 3, 4}. A function is defined from X to N as . Then find the range of f. Q5. If f(x) = ax + b and g(x) = cx + d, then show that f[g(x)] β g[f(x)] is equivalent to f(d) β g(b). ππ₯+π Q6. Let f (x) = ππ₯+π , then if fof= x, then find d in terms of a. Q7. In the group (Z, *) of all integers, where a * b = a + b + 1 for a, b Z, then what is the inverse of β 2? Q8. If f: R R and g: R R defined by f(x) =2x + 3 and g(x) = of x for which f (g(x)) =25. + 7, then find the value Q9. Find the Total number of equivalence relations defined in the set: S = {a, b, c}. Q10. Find whether the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive,symmetric or transitive. Q11. Show that the functionπ: π β π, given by f (x) = 2x, is one-one but not onto. Q12. Find gof and fog, if π: π β π and π: π β π are given by f (x) = cos x and g (x) = 3x2. Q13. Find the number of all one-one functions from set A = {1, 2, 3} to itself. Q14. Let A = {1, 2, 3}. Then find the number of equivalence relations containing (1, 2). Q15. State with reason whether following functions have inverse π: {1,2,3,4} β {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}. 4 Marks Questions 2 Q1. Let π: π β π be a function defined asπ(π₯) = 4π₯ 2 + 12π₯ + 15. Show that π: π β π where, S is the range of f is invertible. Find the inverse of f. Q2. Show that the Relation R in the set π΄ = {π₯ π π§ βΆ 0 β€ π₯ β€ 12}is an equivalence relation. π = {(π, π): |π β π|} is a multiple of 4. π Q3. Show that the function π: πΉ β πΉ given by π(π) = { π βπ one nor onto. ππ π > π ππ π = π ππ π < π is neither one- π₯β2 Q4. If π΄ = π β {3} and π΅ = π β {1}, consider the function π: π΄ β π΅ defined by π(π₯) = π₯β3 . Is f one- one and onto? Justify your answer. Q5. Let π: π β π , π: π β π be two functions given by f(x) = 2x - 3, g (x) = x3 + 5. Find fog-1(x) Q6. Check the injectivity and surjectivity of the following: (i) π: π β π given by π(π₯) = π₯ 2 (ii) π: π β π given by π(π₯) = π₯ 2 . Q7. Determine whether the following relations are reflexive, symmetric, and transitive if relation R,in the set N of Natural numbers is defined as Q8. Consider the binary operation on the set {1,2,3,4,5} defined by Write the operation table of the operation . . . Q9. Let the * be the binary operation on N be defined by H.C.F of a and b. Is * commutative? Is * associative? Does there exist identity for this operation on N? Q10. Let A={-1,0,1,2}, B={-4,-2,0,2} and π, π: π΄ β π΅ be function defined by π(π₯) = 1 π₯ 2 β π₯, π₯ π π΄ and π(π₯) = 2 |π₯ β 2| β 1, π₯ π π΄ . Then, are f and g equal? Justify your answer. Q11. Let f,g and h be functions from π β π . Then show that (i) (ii) Q12. If π: π β π be a function defined by f (x) = 4x3β7. Then show that f is bijection. 3 x Q13. Show that π: [β1,1] π π , given by π (π₯) = x+2 is one-one. Find the inverse of the function f : [β f. Q14. Let N be the set of all natural numbers. R be the relation on N X N defined by (a,b) R (c,d) iff ad = bc β π, π, π, π π π . Show that R is equivalence. Q15. Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R1 be a relation in X given by R1 = {(x, y): x β y is divisible by 3} and R2 be another relation on X given by R2 = {(x, y): {x, y} ο {x, y} ο{2, 5, 8} or {x, y} ο 6 Marks Questions Q1. Let N be the set of all natural numbers. R be the relation on N X N defined by (a,b) R (c,d) iff ad = bc β π, π, π, π π π . Show that R is Equivalence relation. Q2. Q3. Is the function one-one onto Q4. A function f over the set of real numbers is defined as: π(π₯) = 2π₯ + 1 0β€π₯<2 { . π₯β2 2β€π₯β€5 Find whether the function is one-one or onto. 4π₯+3 2 Q5. If π(π₯) = 6π₯β4 , Show thatπππ(π₯) = π₯ for all π₯ β 3. What is the inverse of f(x)? 4 Q6. Define a binary operation * on the set {0,1,2,3,4,5}as π β π = π+π ππ π + π < 6 { π + π β 6 ππ π + π β₯ 6 Show that 0 is the identity for this operation and each element of the set is invertible with 6 β a being the inverse of a. Answers: Functions & Relations 1 Marks Questions Q1. Universal Relation Q2. Not a function Q3. one-one into mapping Q4. {1, 2, 6, 24} Q6. d = β a Q7. 0 Q8. X= +_ 2 Q9. 5 4 Marks Questions Q1. πππ(π) = π[π(π)] = π [ βπβπ β π π ] 6 marks Questions Q5. f-1 (x)= (4y+3)/(6y-4) Q6. is the identify element if