Graphics Reference
In-Depth Information
Proof.
Straightforward.
Definition.
Let
S
be an equivalence relation on a set
X
. If x Œ
X
, then
the equiva-
lence class of x
, denoted by [x], is defined by
[]
=Œ
{
}
xy
XS
.
xy
The
quotient space of
X
by
S
, denoted by
X
/
S
, is defined by
=
[]
{
}
XS
xx
Œ
X
.
Definition.
Let
S
be a relation on a set
X
. The equivalence relation on
X
induced by
S
, or the
induced equivalence relation
, is defined to be intersection of all equivalence
relations on
X
that contain
S
.
One can easily show that the equivalence relation induced by a relation is an
equivalence relation and one can think of it as the “smallest” equivalence relation
containing
S
.
Definition.
A well-defined relation
S
between two sets
X
and
Y
is called a
function
or
map
from the domain of
S
to
Y
. A one-to-one function is called an
injective
function or
injection
. An onto function is called a
surjective function
or
surjection
.
We shall use the notation
f:
XY
Æ
to mean that f is a function between sets
X
and
Y
whose domain is
X
and call f a
function from
X
to
Y
. Given such a function, the standard notation for y, given xfy, is
f(x) and one typically uses the notation y = f(x) when talking about f. One sometimes
calls the range of f the points
traced out
by f.
Definition.
Given a set
X
, define the
diagonal map
d:
XXX
Æ
¥
¥◊◊◊¥
X
by
dx
()
=
(
xx
, ,...,
x
)
.
Definition.
Given maps f
i
:
X
i
Æ
Y
i
, define the
product
map
ff
¥
¥◊◊◊¥
n
:
XX
¥
¥◊◊◊¥
X YY
Æ
¥
¥◊◊◊¥
Y
.
12
1
2
n
1 2
n
by
(
ff
¥
¥◊◊◊¥
fxx
)(
,
,...,
x
)
=
(
fxfx
() ()
,
,...,
fx
( )
)
.
12
n
12
n
1122
n
n
