ch of these relations on the set of all functions from Z to Z are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) { ( f , g ) | f ( 1 ) = g ( 1 ) } b) { ( f , g ) | f ( 0 ) = g ( 0 ) or f ( 1 ) = g ( 1 ) } c) { ( f , g ) | f ( x ) − g ( x ) =1 for all x ∈ Z } d) { ( f , g ) | for some C ∈ Z , for all x ∈ Z , f ( x ) − g ( x ) =C } e) { ( f , g ) | f ( 0 ) = g ( 1 ) and f ( 1 ) = g ( 0 ) }
ch of these relations on the set of all functions from Z to Z are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) { ( f , g ) | f ( 1 ) = g ( 1 ) } b) { ( f , g ) | f ( 0 ) = g ( 0 ) or f ( 1 ) = g ( 1 ) } c) { ( f , g ) | f ( x ) − g ( x ) =1 for all x ∈ Z } d) { ( f , g ) | for some C ∈ Z , for all x ∈ Z , f ( x ) − g ( x ) =C } e) { ( f , g ) | f ( 0 ) = g ( 1 ) and f ( 1 ) = g ( 0 ) }
Solution Summary: The author explains that equivalence relations determine the properties of an equivalent relation that the others lack.
ch of these relations on the set of all functions fromZtoZare equivalence relations? Determine the properties of an equivalence relation that the others lack.
a)
{
(
f
,
g
)
|
f
(
1
)
=
g
(
1
)
}
b)
{
(
f
,
g
)
|
f
(
0
)
=
g
(
0
)
or
f
(
1
)
=
g
(
1
)
}
c)
{
(
f
,
g
)
|
f
(
x
)
−
g
(
x
)
=1 for all
x
∈
Z
}
d)
{
(
f
,
g
)
|
for some C
∈
Z
,
for all
x
∈
Z
,
f
(
x
)
−
g
(
x
)
=C
}
e)
{
(
f
,
g
)
|
f
(
0
)
=
g
(
1
)
and
f
(
1
)
=
g
(
0
)
}
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RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY