Download Relation and Functions

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 𝐴 = {𝑥: −1 ≤ 𝑥 ≤ 1} and S be the subset of Aׂ defined by S = {(x,y) : x2 + y2 = 1}. Is it
a function?
Q3. A mapping 𝑓: 𝑁 → 𝑁 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑓(𝑥) = 2𝑥 ∀𝑥 𝜖 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).
𝑎𝑥+𝑏
Q6. Let f (x) = 𝑐𝑥+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) =
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 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 inverse
𝑓: {1,2,3,4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}.
4 Marks Questions
2
Q1. Let 𝑓: 𝑁 → 𝑅 be a function defined as𝑓(𝑥) = 4𝑥 2 + 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 𝜖 𝑧 ∶ 0 ≤ 𝑥 ≤ 12}is an equivalence relation.
𝑅 = {(𝑎, 𝑏): |𝑎 − 𝑏|} is a multiple of 4.
𝟏
Q3. Show that the function 𝒇: 𝑹 → 𝑹 given by 𝒇(𝒙) = { 𝟎
−𝟏
one nor onto.
𝒊𝒇 𝒙 > 𝟎
𝒊𝒇 𝒙 = 𝟎
𝒊𝒇 𝒙 < 𝟎
is neither one-
𝑥−2
Q4. If 𝐴 = 𝑅 − {3} and 𝐵 = 𝑅 − {1}, consider the function 𝑓: 𝐴 → 𝐵 defined by 𝑓(𝑥) = 𝑥−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 𝑓(𝑥) = 𝑥 2
(ii) 𝑓: 𝑅 → 𝑅 given by 𝑓(𝑥) = 𝑥 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 𝑓, 𝑔: 𝐴 → 𝐵 be function defined by 𝑓(𝑥) =
1
𝑥 2 − 𝑥, 𝑥 𝜖 𝐴 and 𝑔(𝑥) = 2 |𝑥 − 2| − 1, 𝑥 𝜖 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.
3
x
Q13. Show that 𝑓: [−1,1] 𝜖 𝑅, given by 𝑓 (𝑥) = x+2 is one-one. Find the inverse of the
function f : [–
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 𝜖 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} 
{x, y} {2, 5, 8} or {x, y} 
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 ∀ 𝑎, 𝑏, 𝑐, 𝑑 𝜖 𝑁 . 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: 𝑓(𝑥) =
2𝑥 + 1
0≤𝑥<2
{
.
𝑥−2
2≤𝑥≤5
Find whether the function is one-one or onto.
4𝑥+3
2
Q5. If 𝑓(𝑥) = 6𝑥−4 , Show that𝑓𝑜𝑓(𝑥) = 𝑥 for all 𝑥 ≠ 3. What is the inverse of f(x)?
4
Q6. Define a binary operation * on the set {0,1,2,3,4,5}as 𝑎 ∗ 𝑏 =
𝑎+𝑏
𝑖𝑓 𝑎 + 𝑏 < 6
{
𝑎 + 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 & Relations
1 Marks Questions
Q1. Universal Relation
Q2. Not a function
Q3. one-one into mapping
Q4. {1, 2, 6, 24}
Q6. d = – a
Q7. 0
Q8. X= +_ 2
Q9. 5
4 Marks Questions
Q1. 𝒇𝒐𝒈(𝒚) = 𝒇[𝒈(𝒚)] = 𝒇 [
√𝒚−𝟔 − 𝟑
𝟐
]
6 marks Questions
Q5. f-1 (x)= (4y+3)/(6y-4)
Q6.
is the identify element if