Mappings
Definition. A mapping is a relation
from
to
with the property that for every
there is a unique
such that
, in which case we write
. Mappings are sometimes called maps or functions; and we shall generally reserve the latter name for the case when
is
or some subset thereof.
If and
are mappings, we denote by
their composition:
,
.
If and
, we define the image of
and the inverse image of
under the mapping
by:
,
.
The map defined by the second formula commutes with union, intersections, and complements:
,
,
.
If is a mapping,
is called the domain of
and
is called the range of
.
is said to be injective if
only when
, surjective if
, and bijective if it is both injective and surjective. If
is bijective, then it has an inverse
such that
and $f\circ f^{-1}$ are the identity mappings on
and
, respectively. If
, we denote by
the restriction of
to
:
,
for
.
A sequence in a set is a mapping from
into
.
A finite sequence is a map from into
where
.
If is a sequence and
satisfies
whenever
, the composition
is called a subsequence of $f$.
If is an indexed family of sets, their Cartesian product
is the set of all maps
such that
for every $\alpha\in A$.
If and
, we define the
th projection or coordinate map
by
. We also frequently write
and
instead of
and
and call
the
th coordinate of
.
If the sets are all equal to some fixed set
, we denote
by
:
the set of all mappings from
to
.
If ,
is denoted by
and may be identified with the set of ordered
-tuples of elements of
.
“To let go does not mean to stop caring, it means I can’t do it for someone else. To let go is not to cut myself off, it’s the realization I can’t control another. To let go is not to enable, but allow learning from natural consequences. To let go is to admit powerlessness, which means the outcome is not in my hands. To let go is not to try to change or blame another, it’s to make the most of myself. To let go is not to care for, but to care about. To let go is not to fix, but to be supportive. To let go is not to judge, but to allow another to be a human being. To let go is not to be in the middle arranging all the outcomes, but to allow others to affect their destinies. To let go is not to be protective, it’s to permit another to face reality. To let go is not to deny, but to accept. To let go is not to nag, scold or argue, but instead to search out my own shortcomings and correct them. To let go is not to adjust everything to my desires, but to take each day as it comes and cherish myself in it. To let go is not to criticize or regulate anybody, but to try to become what I dream I can be. To let go is not to regret the past, but to grow and live for the future. To let go is to fear less and LOVE more. Remember: the time to love is short.” ~ Author Unknown
To be continued tomorrow 🙂
References:
[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 3-4.
[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf