# What Exactly Is The General Concept Of A Mathematical “Mapping”? How Are The Mappings Used?

I am seeking to understand the concept of a "mapping" in Mathematics.

I tried reading "pure" mathematical information, and I encountered the term "mapping." My interpretation of the term is that it is a relationship between distinct sets.

The examples I saw focused entirely on functions.

I also created this function example of my attempt to understand the concept:

$N : \{e | e > 100 \land e < 1000\} \to \{(e * 5)\}$

I apologize for any scattered notation.

My attempt was intended to operate like this:

Each element supplied to the '$N$' function was intended to be separately and orderly processed by the function. Values not accepted by the set-builder notation in the domain set, such as ($3000$), would be disregarded. The element would then be multiplied by five, and added into the range set.

However, I am still confused on how a mapping differs from a regular function. Also, what is the significance of the mappings in pure Mathematics?

I have very little knowledge of the subject, so I apologize if I misinterpret any explanations provided.

• In many contexts, function and mapping denote the same thing. – masterxilo Oct 4 '16 at 16:21
• The notation you are looking for is $N : (100, 1000) \to \mathbb{R}, e \mapsto 5e$. – Anon Oct 4 '16 at 16:26
• A related question math.stackexchange.com/questions/95741/… – rtybase Oct 4 '16 at 16:55

We call $f$ a mapping if it is a way to uniquely assign an output to a given input.
I can write: $f: 0 \mapsto 1, 1 \mapsto 0$ to say that $f$ is the mapping that assigns to the input $0$ the output $1$ and to $1$ the output $0$. We say $f$ maps 0 to 1 and 1 to 0.
There are lots of other notations for this, but the idea is always that for any $a$ and $b$, we can say for any mapping $f$ whether $f$ maps $a$ to $b$ or not. Furthermore, for a mapping $f$ and an $a$, there is at most one $b$ such that $f$ maps $a$ to $b$.
You are almost right, a map "is a relationship between distinct sets". Almost, because it is a relation between two sets A and B (particularly $A=B$) which associates every element $x \in A$ to a unique (!!!) element $y \in B$. Map is a function, these two terms are synonymous, assuming map is not confused with relation like "many to many mapping" in databases which in fact should be "many to many relation".
For example $A=\{0,1,3\}$ and $B=\{1,2\}$. $R=\{(0,1), (0,2), (1,1)\}$ is a relation, but not a function/map because $0 \in A$ is associated to $1 \in B$ and $2 \in B$.
Another example, circle $B_r(0,0)$ (circle of radius $r$ with origin in $(0,0)$), $B_r(0,0) = \{(x,y) \in \mathbb{R}^2| x^2+y^2=r^2\}$ is a relation, but not a function/map because, as an example, $x=0$ is associated to $y=r$ and $y=-r$.