Chapter - 1 ( Relations and functions )
Chapter-1
Relation and functions
Topic:Relations
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1) Show
that the relation R on the set R of real numbers, defined as
R = {(a, b): a ≤ b2} is neither
reflexive nor symmetric nor transitive.
2)
Check whether the relation R in R defined by R = {(a, b)
: a ≤ b3} is
reflexive, symmetric or transitive.
3)
Show 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. Show that all the
elements of {1, 3, 5} are related to each
other and all the elements of {2, 4}
are related to each other. But no element
of {1, 3, 5} is related to any
element of {2, 4}.
4)
Show that each of the relation R in the set A = {x ∈
Z
:
0 ≤ x ≤
12},
given by
(i) R = {(a, b)
:|a – b| is a multiple of 4}
(ii) R = {(a, b)
: a = b}
is an equivalence relation. Find the set of
all elements related to 1 in each case.
5)
If R = {(x, y): x + 2y = 8} is a relation on
N, write the range of R.
6)
State the reason for the relation R in the set {1, 2, 3} given by
R = {(1, 2), (2, 1)} not to be
transitive.
7)
Let R = {(a, a3) : a is a prime number
less than 5} be a relation.
Find the range of R.
8)
Check whether the relation R defined on the set A = {1, 2, 3, 4,
5, 6} as
R = {(a, b) : b =
a + 1} is reflexive, symmetric or transitive.
9)
Show that the relation S in the set A = {x ∈
Z
:
0 ≤ x ≤
12}
given by
S = {(a, b) : a,
b ∈ Z, |a – b|
is divisible by 3} is an equivalence relation.
10)
Check whether the relation R defined on the set A = {1, 2, 3, 4,
5, 6} as
R = {(a, b) : b =
a + 1} is reflexive, symmetric or transitive.
11)
Show that the relation R on the set Z of all integers, given by
R = {(a, b) : 2
divides (a – b)} is an equivalence relation.
12)
Show that the relation S in the set A = {x ∈
Z
:
0 ≤ x ≤
12}
given by
S = {(a, b) : a,
b ∈ Z, |a – b|
is divisible by 3} is an equivalence relation.
13)Determine
whether the relation R defined on the set ℝ of all real numbers as
R = {(a, b) : a,
b ∈ ℝ
and a – b +
numbers}, is reflexive, symmetric and transitive.
14) State
the reason for the relation R in the set {1, 2, 3} given by
R = {(1, 2),(2, 1)} not to be transitive.
15) Let
A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2,
5), (3, 6)} be a
function from A to B. State whether f is one-one or not.
16) Let A = {x
∈ Z
:
0 ≤ x ≤
12}.
Show that R = {(a, b)
: a, b ∈ A, |a – b|
is divisible by 4} is an equivalence
relation. Find the set of all elements
related to 1. Also write the equivalence
class [2].
17) Show that the
relation R defined in the set A of all triangles as
R = {(T1, T2) : T1
is similar to T2}, is equivalence relation. Consider three
right angle triangles T1 with
sides 3, 4, 5, T2 with sides 5, 12, 13 and T3
with sides 6, 8, 10. Which triangles
among T1, T2 and T3 are related?
18) Prove that the greatest
integer function f: R->R
given by f(x) = [x], is
neither one-one nor onto, where [x] denotes the greatest integer less
than or
equal to x.
19) Let N denote the set of all natural
numbers and R be the relation
on N × N
defined by (a, b) R (c, d) if ad(b + c) = bc(a + d). Show that R is an
equivalence relation.
20) Let Z be the set of all integers and R be 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.
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