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Byju's Answer
Standard XII
Mathematics
De Morgan's Law
nbsp;Prove th...
Question
Prove that the relation R on set Nx N (a,b) R(c,d) which implies a+d=b+c is an equivalence relation
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Solution
Dear student
We
observe
the
following
properties
of
relation
R
:
Reflexivity
:
Let
(
a
,
b
)
be
an
arbitrary
element
of
N
×
N
.
Then
,
(
a
,
b
)
∈
N
×
N
⇒
a
,
b
∈
N
⇒
a
+
b
=
b
+
a
[
By
commutativity
of
add
.
on
N
]
⇒
(
a
,
b
)
R
(
a
,
b
)
Thus
,
(
a
,
b
)
R
(
a
,
b
)
for
all
(
a
,
b
)
∈
N
×
N
.
So
,
R
is
reflexive
on
N
×
N
.
Symmetry
:
Let
(
a
,
b
)
,
(
c
,
d
)
∈
N
×
N
be
such
that
(
a
,
b
)
R
c
,
d
.
Then
,
(
a
,
b
)
R
(
c
,
d
)
⇒
a
+
d
=
b
+
c
⇒
c
+
d
=
d
+
a
[
By
commutativity
of
add
.
on
N
]
⇒
(
c
,
d
)
R
(
a
,
b
)
[
By
defn
.
of
R
]
Thus
,
(
a
,
b
)
R
(
c
,
d
)
⇒
(
c
,
d
)
R
(
a
,
b
)
for
all
(
a
,
b
)
,
(
c
,
d
)
∈
N
×
N
.
So
,
R
is
symmetric
on
N
×
N
.
Transitivity
:
Let
(
a
,
b
)
,
(
c
,
d
)
,
(
e
,
f
)
∈
N
×
N
such
that
(
a
,
b
)
R
(
c
,
d
)
and
(
c
,
d
)
R
(
e
,
f
)
.
Then
,
a
,
b
R
c
,
d
⇒
a
+
d
=
b
+
c
(
c
,
d
)
R
(
e
,
f
)
⇒
c
+
f
=
d
+
e
⇒
a
+
d
+
c
+
f
=
(
b
+
c
)
+
(
d
+
e
)
⇒
a
+
f
=
b
+
e
⇒
(
a
,
b
)
R
(
e
,
f
)
Thus
,
(
a
,
b
)
R
(
c
,
d
)
and
(
c
,
d
)
R
(
e
,
f
)
⇒
(
a
,
b
)
R
(
e
,
f
)
for
all
(
a
,
b
)
,
(
c
,
d
)
,
(
e
,
f
)
∈
N
×
N
.
So
,
R
is
transitive
on
N
×
N
.
Hence
,
R
being
reflexive
,
symmetric
and
transitive
,
is
an
equivalence
relation
on
N
×
N
.
Regards
Suggest Corrections
0
Similar questions
Q.
Show that the relation R on the set N×N
(a,b)R(c,d) if and only if a+d = b+c is an equivalence relation
Q.
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
a
d
(
b
+
c
)
=
b
c
(
a
+
d
)
,
then show that
R
is an equivalence relation.
Q.
Let
N
denote the set of natural numbers and
R
be a relation on
N
×
N
defined by
(
a
,
b
)
R
(
c
,
d
)
⟺
a
d
(
b
+
c
)
=
b
c
(
a
+
d
)
.
Then on
N
×
N
,
R
is
Q.
Let
N
denote the set of all natural numbers and
R
be the relation on
N
×
N
defined by
(
a
,
b
)
R
(
c
,
d
)
⟺
a
d
(
b
+
c
)
=
b
c
(
a
+
d
)
. Check weather
R
is an equivalence relation.
Q.
Let N denotes the set of natural numbers and R is a relation in N
×
N. which of the following is not an equivalence relation in N
×
N?
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