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Relations and Functions
NCERT Solutions
NCERT Solutions
Relations and Functions
4 Solutions
Q1
EXERCISE 1.1
Determine whether each of the following relations are reflexive, symmetric and transitive:
(i)
Relation R in the set
A
=
{
1
,
2
,
3
,
…
,
13
,
14
}
\mathrm{A}=\{1,2,3, \ldots, 13,14\}
A
=
{
1
,
2
,
3
,
…
,
13
,
14
}
defined as
R
=
{
(
x
,
y
)
:
3
x
−
y
=
0
}
\mathrm{R}=\{(x, y): 3 x-y=0\}
R
=
{(
x
,
y
)
:
3
x
−
y
=
0
}
(ii)
Relation R in the set
N
\mathbf{N}
N
of natural numbers defined as
R
=
{
(
x
,
y
)
:
y
=
x
+
5
and
x
<
4
}
\mathrm{R}=\{(x, y): y=x+5 \text { and } x<4\}
R
=
{(
x
,
y
)
:
y
=
x
+
5
and
x
<
4
}
(iii)
Relation R in the set
A
=
{
1
,
2
,
3
,
4
,
5
,
6
}
\mathrm{A}=\{1,2,3,4,5,6\}
A
=
{
1
,
2
,
3
,
4
,
5
,
6
}
as
R
=
{
(
x
,
y
)
:
y
is divisible by
x
}
\mathrm{R}=\{(x, y): y \text { is divisible by } x\}
R
=
{(
x
,
y
)
:
y
is divisible by
x
}
(iv)
Relation R in the set
Z
\mathbf{Z}
Z
of all integers defined as
R
=
{
(
x
,
y
)
:
x
−
y
is an integer
}
\mathrm{R}=\{(x, y): x-y \text { is an integer }\}
R
=
{(
x
,
y
)
:
x
−
y
is an integer
}
(v)
Relation R in the set A of human beings in a town at a particular time given by
(a)
R
=
{
(
x
,
y
)
:
x
\mathrm{R}=\{(x, y): x
R
=
{(
x
,
y
)
:
x
and
y
y
y
work at the same place
}
\}
}
(b)
R
=
{
(
x
,
y
)
:
x
\mathrm{R}=\{(x, y): x
R
=
{(
x
,
y
)
:
x
and
y
y
y
live in the same locality
}
\}
}
(c)
R
=
{
(
x
,
y
)
:
x
\mathrm{R}=\{(x, y): x
R
=
{(
x
,
y
)
:
x
is exactly 7 cm taller than
y
}
y\}
y
}
(d)
R
=
{
(
x
,
y
)
:
x
\mathrm{R}=\{(x, y): x
R
=
{(
x
,
y
)
:
x
is wife of
y
}
y\}
y
}
(e)
R
=
{
(
x
,
y
)
:
x
\mathrm{R}=\{(x, y): x
R
=
{(
x
,
y
)
:
x
is father of
y
}
y\}
y
}
Q2
EXERCISE 1.1
Show that the relation R in the set
R
\mathbf{R}
R
of real numbers, defined as
R
=
{
(
a
,
b
)
:
a
≤
b
2
}
\mathrm{R}=\left\{(a, b): a \leq b^{2}\right\}
R
=
{
(
a
,
b
)
:
a
≤
b
2
}
is neither reflexive nor symmetric nor transitive.
Q3
EXERCISE 1.1
Check whether the relation R defined in the set
{
1
,
2
,
3
,
4
,
5
,
6
}
\{1,2,3,4,5,6\}
{
1
,
2
,
3
,
4
,
5
,
6
}
as
R
=
{
(
a
,
b
)
:
b
=
a
+
1
}
\mathrm{R}=\{(a, b): b=a+1\}
R
=
{(
a
,
b
)
:
b
=
a
+
1
}
is reflexive, symmetric or transitive.
Q4
EXERCISE 1.1
Show that the relation R in
R
\mathbf{R}
R
defined as
R
=
{
(
a
,
b
)
:
a
≤
b
}
\mathrm{R}=\{(a, b): a \leq b\}
R
=
{(
a
,
b
)
:
a
≤
b
}
, is reflexive and transitive but not symmetric.
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