Download Relations and Functions

Relations and Functions
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1.
Show that the relation ‘>’ on the set R of real numbers is transitive, but is neither
reflexive nor symmetric.
2.
Show that the relation ‘is the square of’ on the set of natural numbers satisfies none of the
properties; reflexivity, symmetry, transitivity.
3.
Let A = {1, 2, 3} and R = {(1, 2) (1, 1) (2, 3)} be a relation on A. What minimum number
of elements may be adjoined with the elements of R so that it may become transitive.
[Ans: 1 i.e.(1, 3)]
4.
How many relations are possible from a set A of m elements to another set B of n
elements? Why ?
[Ans : 2mn]
5.
Let A = {1, -1, 2, 3} and B={1, 4, 9, 16}
Show that the function (x) = x2 is many one and onto.
6.
Let RR be defined as follows:
(x) =
1 if x  Q
-1 if x  Q
 
1
Find (a) f  , f , f 2
(b) Range of f (c) (c) Pre-image of 1 and -1.
2
Ans : (a) +1, -1, -1
b) {1, -1}
c) Set of Rationals; Irrationals
7.
8.
9.
Let RR be such that f(x) = 2x. Determine
a) Range of 
b) {x if (x) = 1}
c) Whether (x+y)(x)(y) holds?
Ans : a) R+
b) {0}
c) Holds.
Let A = {1, 2, 3, 5}, let  = {(1, 5) (2, 1) (3, 3) (5, 2)} and
g = {(1, 3), (2, 1), (3, 2), (5, 5)  Find (i) og (ii) go (iii) o and (iv) gog
Ans : (i) {(1, 3), (2, 5), (3, 1), (5, 2)}
(ii) {(1, 5), (2, 3), (3, 2), (5, 1)}
(iii) {(1, 2), (2, 5), (3, 3), (5, 1)}
(iv) {(1, 2), (2, 3), (3, 1), (5, 5)}
Explain why the following functions XY do not have inverses;
1
(i)
X = QY Qand f(x) = for all x  X
x
(ii)
X = {1, 2, 3, 4, 5}, Y ={0, 1} and f(x) = 0 for x = 1, 2, 3, 4 and f(5) = 1
(iii)
X = R = Y and f(x) = x2 for all x  R
(iv)
X = R, Y = {x : x R, x  0 } and f(x) = x2 for all x  R
Ans : (i) not onto (ii) not one-one (iii) neither onto nor one-one (iv) not 1-1
10.
Let RR be a function given by (x) = ax + b for x  R
Find the constants a and b such that o = IR
Ans : a  1, b = 0, any real number.
11.
Examine the nature of each of the functions
(i) RR ; (x) = x3 for all x  R 
(ii) gNN ; g(x) = 2x + 3 for all x  N 
(iii) hRR ; h(x) = x + 1 for all x  R 
(iv) kCC ; k(x) = x3 for all x  C 
Ans : (i)one-one, onto; (ii) one-one, into (iii) one-one, onto (iv) many one onto
12.
Let A = {-2, -1, 0, 1, 2} and AZ, given by (x) = x2 - 2x -3
Find a) Range of  b) pre-image of 6, -3, 5.
Ans : a) Range () = {0, 5, -3, -4} b) 0; 0, 2; -2
13.
Show that the function RR , (x) = Sin x, for all x  R is neither one-one nor onto.
14.
Is (x) =
15.
Let the binary operation ‘o’ be defined on Z by lom = l - m + lm for all l, m  Z. Prove
that the composition is neither commutative nor associative.
16.
Let A be the set of all real numbers except -1 and an operation ‘o’ be defined by aob = a
+ b + ab for all a, b  A. Prove that
(i) A is closed under the given operation
(ii) the given operation is commutative as well as associative.
(iii) the number o is an identity element.
a
(iv) every element a of A has
as its inverse.
1 a
Consider the operation ‘  ’ and  on the set R for all real numbers defined as
a  b = a  b , and a  b = a for all a, b  R Prove that ‘  ’ is commutative but not
17.
18.
x
invertible or its domain? If so find . Also verify that o-1 = x.
x 1
x
Ans : Yes, (x) =
, o-1(x) = x
x 1
associative, but  is associative and not commutative, further prove that  is distributive
over , i.e. a  (b  c) = (a  b)  (a  c) and (a  b)  c = (a  c)  (b  c) for a, b E  R.
Let a binary ‘  ’ on R be defined by a  b = a b for all a, b  R. Prove that
(i) R is closed under the given operation.
(ii) the given operation is not commutative
(iii) the given operation is associative
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28.
Let the binary operation  on the set N of natural numbers be defined as
m  n = gcd(m,n) for m, n  N. Determine whether the operation is commutative,
associative.
Ans : is both
If f(x) = x + 7, g(x) = x-7, x  R find fog(7). (cbse 2008)
Is the binary operation * defined on the set N/Q, given by a*b=a+b/2 for all a, b
belonging to N/Q commutative? Assosiative?
(CBSE 2008)
Prove 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. (CBSE 2009)
If the binary operation * on the set of integers Z , is defined by a*b= a  3b 2 , then find
the value of 2*4
(CBSE 2009)
What is the range of the function f ( x) 
x 1
? (CBSE 2010)
x 1
Let Z be the set of all integers and R be the relation on Z defined as
R  a, b : a, b  Z , and a  b.....is.....divisible.....by.....5. Prove that R is an
equivalence relation. (CBSE 2010)
Show that the relation S in the set R of real numbers, defined as
S  a, b : a, b  R....and .....a  b 3 is neither reflexive, nor symmetric nor transitive.
(CBSE 2010
State the reason for the relation R in th{1,2,3} given by R = {(1,2), (2,1)} not to be
transitive. (CBSE 2011)


Consider the binary operation * on the set {1,2,3,4,5} defined by a * b = min {a,b}.
Write the operation table of the operation *.(CBSE 2011)
Sol:
*
1
2
3
4
5
1
1
1
1
1
1
2
1
2
2
2
2
3
1
2
3
3
3
4
1
2
3
4
4
5
1
2
3
4
5