Relation and Functions



Chapter 1 Relation and Functions1 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 A=x:-1≤x≤1 and S be the subset of A? defined by S = {(x,y) : x2 + y2 = 1}. Is it a function?Q3. A mapping f:N→N is defined by fx=2x ?x ? N , 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)] – g[f(x)] is equivalent to f(d) – g(b).Q6. Let f (x) = ax+bcx+d , 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 – 2?Q8. If f: R R and g: R R defined by f(x) =2x + 3 and g(x) = + 7, then find the value of x for which f (g(x)) =25.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 functionf:N→N, given by f (x) = 2x, is one-one but not onto.Q12. Find gof and fog, if f:R→R and g:R→R 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 inversef:1,2,3,4→10 with f = {(1, 10), (2, 10), (3, 10), (4, 10)}. 4 Marks QuestionsQ1. Let f:N→R be a function defined asfx=4x2+12x+15. Show that f:N→S?where, S is the range of f is invertible. Find the inverse of f.Q2. Show that the Relation R in the set A=x ? z :0≤x≤12is an equivalence relation. R=a,b:a-b is a multiple of 4. Q3. Show that the? function f:R→R given by fx=1 if x >00 if x=0-1 if x<0 is neither one-one nor onto.Q4. If A=R-3 and B=R-1, consider the function f:A→B defined by fx=x-2x-3 . Is f ?one- one and onto? Justify your answer.Q5. Let f:R→R, g:R→R 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)? f:N→N given by fx=x2? (ii) f:R→R given by fx=x2?.??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 f,g:A→B be function defined? by fx= x2-x, x ? A?and gx=2x-12-1, x ? A . Then, are f and g ?equal? Justify your answer.Q11. Let f,g and h be functions from R→R. Then show that ? (i) ?????? ? (ii)?????? Q12. If f:R→R be a function defined by f (x) = 4x3–7. Then show that f is bijection.Q13. Show that f:-1,1 ? R, given by f (x) =xx+2 is one-one. Find the inverse of the function f : [–1, 1] ? & Range 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 ? a,b,c,d ? N . 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 – y is divisible by 3} and R2 be another relation on X given by R2 = {(x, y): {x, y} ?{1, 4, 7}} or {x, y} {2, 5, 8} or {x, y} ?{3, 6, 9}}. Show that R1 = R2.6 Marks QuestionsQ1. 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 ? a,b,c,d ? N . Show that R is Equivalence relation.Q2. Q3. Is the function one-one ontoQ4. A function f over the set of real numbers is defined as: fx=2x+1 0≤x<2x-2 2≤x≤5. Find whether the function is one-one or onto.Q5. If fx=4x+36x-4 , Show thatfofx=x for all x≠23. What is the inverse of f(x)?Q6. Define a binary operation * on the set {0,1,2,3,4,5}as a*b= a+b if a+b <6a+b-6 if a+b ≥6 Show that 0 is the identity for this operation and each element of the set is invertible with 6 – a ?being the inverse of ?a.?Answers: Functions & Relations1 Marks QuestionsQ1. Universal RelationQ2. Not a functionQ3. one-one into mappingQ4. {1, 2, 6, 24} Q6. d = – a Q7. 0Q8. X= +_ 2Q9. 54 Marks QuestionsQ1. fogy=fgy=fy-6 - 326 marks QuestionsQ5. f-1 (x)= (4y+3)/(6y-4)Q6. is the identify element if ................
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