#### Mappings

Definition. A mapping $f: X\to Y$ is a relation $R$ from $X$ to $Y$ with the property that for every $x\in X$ there is a unique $y\in Y$ such that $xRy$, in which case we write $y=f(x)$. Mappings are sometimes called maps or functions; and we shall generally reserve the latter name for the case when $Y$ is $\mathbb C$ or some subset thereof.

If $f: X\to Y$ and $g: Y\to Z$ are mappings, we denote by $g\circ f$ their composition:

$g\circ f: X\to Z$, $g\circ f(x)=g(f(x))$.

If $D\subset X$ and $E\subset Y$, we define the image of $D$ and the inverse image of $E$ under the mapping $f: X\to Y$ by:

$f(D)=\{f(x): x\in D\}$, $f^{-1}(E)=\{x: f(x)\in E\}$.

The map $f^{-1}: \mathcal P(Y)\to\mathcal P(X)$ defined by the second formula commutes with union, intersections, and complements:

$f^{-1}(\cup_{\alpha\in A}E_\alpha)=\cup_{\alpha\in A}f^{-1}(E_\alpha)$,

$f^{-1}(\cap_{\alpha\in A}E_\alpha)=\cap_{\alpha\in A}f^{-1}(E_\alpha)$,

$f^{-1}(E^c)=(f^{-1}(E))^c$.

If $f: X\to Y$ is a mapping, $X$ is called the domain of $f$ and $f(X)$ is called the range of $f$. $f$ is said to be injective if $f(x_1)=f(x_2)$ only when $x_1=x_2$surjective if $f(X)=Y$, and bijective if it is both injective and surjective. If $f$ is bijective, then it has an inverse $f^{-1}: Y\to X$ such that $f^{-1}\circ f$ and $f\circ f^{-1}$ are the identity mappings on $X$ and $Y$, respectively. If $A\subset X$, we denote by $f|A$ the restriction of $f$ to $A$:

$(f|A): A\to Y$, $(f|A)(x)=f(x)$ for $x\in A$.

A sequence in a set $X$ is a mapping from $\mathbb N$ into $X$.

A finite sequence is a map from $\{1, 2, ..., n\}$ into $X$ where $n\in \mathbb N$.

If $f:\mathbb N\to X$ is a sequence and $g: \mathbb N\to\mathbb N$ satisfies $g(n) whenever $n, the composition $f\circ g$ is called a subsequence of $f$.

If $\{X_\alpha\}_{\alpha\in A}$ is an indexed family of sets, their Cartesian product $\prod_{\alpha\in A} X_\alpha$ is the set of all maps $f: A\to\cup_{\alpha\in A}X_\alpha$ such that $f(\alpha)\in X_\alpha$ for every $\alpha\in A$.

If $X=\prod_{\alpha\in A}X_\alpha$ and $\alpha\in A$, we define the $\alpha$th projection or coordinate map $\pi_\alpha: X\to X_\alpha$ by $\pi_\alpha(f)=f(\alpha)$. We also frequently write $x$ and $x_\alpha$ instead of $f$ and $f(\alpha)$ and call $x_\alpha$ the $\alpha$th coordinate of $x$.

If the sets $X_\alpha$ are all equal to some fixed set $Y$, we denote $\prod_{\alpha\in A}X_\alpha$ by $Y^A$:

$Y^A=$ the set of all mappings from $A$ to $Y$.

If $A=\{1, 2, ..., n\}$, $Y^A$ is denoted by $Y^n$ and may be identified with the set of ordered $n$-tuples of elements of $Y$.

“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