RELATIONS AND FUNCTIONS
19
UNSOLVED PROBLEMS EXERCISE – I Q.1
In the set N of all natural numbers, let a relation R be defined by R = {(x, y) : x N, y N, x – y is divisible by 5} prove that R is an equivalence relation.
Q.2
Let ‘m’ be a given positive integer. Prove that the relation, ‘congruence modulo m’ on the set z of all integers defined by a b (mod m) (a – b) is divisible by m is an equivalence relation.
Q.3
Prove that the relation R in the set of integers Z defined by xRy xy = yx x, y Z is an equivalence relation.
Q.4
Let N be the set of all natural numbers and R be the relation on N × N defined by (i)
(a, b) R (c, d) ad = bc
(ii)
(a, b) R (c, d) ad (b + c) = bc (a + d) prove that R. is an equivalence relation in each case.
Q.5
Q.6
If f : X Y and A, B X, then prove that (i)
f(A B) = f(A) f(B)
(i)
f(A B) f(A) f(B)
If f : X Y and A, B Y, then prove that (i)
f–1 (A B) = f–1 (A) f–1 (B)
(ii)
f–1 (A B) = f–1 (A) f–1 (B)
(iii)
f–1 (A – B) = f–1 (A) – f–1 (B)
Q.7
Prove that only one-one onto function has inverse function.
Q.8
Prove that the product of any function with the identity function is the function itself.
Q.9
Prove that the product of any invertible function f with its inverse f –1 is an identity function.
Q.10
Prove that composite of functions is associative.
Q.11
Let f : A B and g : B A such that gof is an identity function on A and fog is an identity function on B. Then, g = f–1
Q.12
Let f : A B and g : B C be one-one onto functions. Then gof is also one-one onto and (gof) –1 = f–1og–1
Q.13
Let f : N N be defined by
n 1, if n is odd f(x) = n 1, if n is even show that f is a bijection. Q.14
Show that the function f : R – {3} R – {1} given by f(x) =
x2 is a bijection. x3
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RELATIONS AND FUNCTIONS
20 Q.15
Q.16
(i)
If f : A B and g : B C are one-one functions. Show that gof is a one-one function.
(ii)
If f : A B and g : B C are onto functions. Show that gof is an onto function.
Prove that the composition of two bijections is a bijection i.e. if f and g are two bijections, then gof is also a bijections.
10 x 10 x
Q.17
If f : R (–1, 1) defined by f(x) =
Q.18
Let S = N × N and ‘*’ be an operation on S defined by (a, b) * (c, d) = (ac, bd) for all a, b, c, d N. Determine
10 x 10 x
is invertible find f–1.
whether ‘*’ is a binary operation on S. If yes, check the commutativity and associativity.
Q.19
Q.20
Let Q be the set of all rational numbers, define an operation on * Q – {–1} by a * b = a + b + ab. show that (i)
‘*’ is a binary operation on Q – {–1}
(ii)
‘*’ is commutative
(iii)
‘*’ is associative
(iv)
zero is the identity element in Q – {–1} for *
(v)
a . where a Q – {–1} a–1 = 1 a
Let S = R0 × R, where Ro denote the set of all non-zero real numbers. A binary operation ‘*’ is defined on s as follows. (a, b) * (c, d) = (ac, bc + d) for all (a, b) (c, d) R0 × R (i)
Find the identity element in S.
(ii)
Find the invertible element in S.
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RELATIONS AND FUNCTIONS
21
BOARD PROBLES EXERCISE – II Q1.
Q2.
Q3.
Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b N is an equivalence relation. (C.B.S.E. 2008) Let T be the set of all triangles in a plane with R as a relation in T given by R = {(T1, T2) : T1 T2}. Show that R is an equivalence relation. (C.B.S.E. 2008) Show that the relation R defined by (a, b) R(c, d) a + d = b + c on the set N × N is an equivalence relation. (C.B.S.E. 2008)
Q4.
Q5.
Q6. Q7. Q8.
Q9.
Q10. Q11.
Q12.
Q13.
(i) Is the binary operation *, defined on the set N, given by a * b =
ab for all a, b Q, commutative ? 2
(ii) Is the above binary operation * associative ? (C.B.S.E. 2008) Let * be a binary operation defined by a * b = 3a + 4b – 2. Find 4 * 5. (C.B.S.E. 2008) 3x 2 If f(x) is an invertible function, find the inverse of f(x) = . 5 (C.B.S.E. 2008) Let * be a binary operation defined by a * b = 2a + b – 3. Find 3 * 4. (C.B.S.E. 2008) If f(x) = x + 7 and g(x) = x – 7, find (fog) (7). (C.B.S.E. 2008)
n 1 2 , if n is odd Let f : N N be defined by f(n) = n for all n N , if n is even 2 Find whether the function f is bijective. (C.B.S.E. 2009) Let * be a binary operation on N given by a * b = HCF (a, b), a, b N. Write the value of 22 * 4. (C.B.S.E. 2009) Show that the relation S defined on the set N × N by (a, b) S (c, d) a + d = b + c is an equivalence relation. (C.B.S.E. 2010) If f : R R be defined by f(x) = (3 – x3)1/3, then find fof(x). (C.B.S.E. 2010) a b, A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as : a * b = a b 6,
if a b 6 if a b 6
Show that zero is the identity for this operation and each element ‘a’ of the set is invertible with 6 – a, being the inverse of ‘a’. (C.B.S.E. 2011) Q14.
Q15.
Q.16
Let f : R R be defined as f(x) = 10x + 7. Find the function g : R R such that gof = fog = IR. (C.B.S.E. 2011)
x 1, if x is odd Show that f : N N, given by f(x) is both one - one and onto : x 1, if x is even (C.B.S.E. 2012) OR Consider the binary operations * : R × R R and o : R × R R defined as a * b = | a – b | and aob = a for all a, b R. Show that ‘*’ is commutative but not associative, ‘o’ is associative but not commutative 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.
[CBSE 2013]
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