**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