What Notations Describe a Mapping How are these notations related (if at all), or are they the same?  Since I'm not sure if it's what people usually write, I'll clarify $\langle,\rangle$ is a mapping. 


*

*$\langle a,b\rangle=c$

*$a:b\to c$

*$a(b)=c$ (a as a function of b)

*$\langle,\rangle:(a,b)\to c$ or $\langle,\rangle:a\times b\to c$


I'm also not sure what any of these mean, so there could be mistakes. Some of the notation I've used here is from a youtube channel XylyXylyX, what is a tensor. 
Edits: I think 2 and 3 are the same, but someone told me 2 and 1 mean the same thing and $a(b)$ is just one type of way to make $a$ be a map (act on something). Or (as in 4) is $\langle,\rangle$ the map but information about it is encoded in $a$ ($a$ is just used somewhere but is part of the function like $\langle a,b\rangle=ab$)? 
Technical changes: I added "if at all" to the first sentence and explicitly stated I'm using $\langle,\rangle$ because that's what I'm used to, even though it's also sometimes used as an inner product. Since this may have made the question misleading I changed the title from "Notations for Mappings" to "What Notations Describe a Mapping". Hopefully these flush out the question a little more. 
 A: The notation $$f:A\to B$$ is meant to say $f$ is a map whose domain is $A$ and whose codomain is $B$. $A$ and $B$ are both sets. E.g. $\text{sq}:\mathbb{R}\to\mathbb{R}$ means $\text{sq}$ is a real valued function defined over the real numbers.
If you have $f:A\to B$, then we also have the notation $$f:a\mapsto b$$ where $a$ is an element of $A$ and $b$ is an element of $B$. This can be used to fix the notation for the evaluation of the map: E.g. $\text{sq}:x\mapsto \text{sq}(x)$. You can also prescript the actual map in that moment, by defining what $\text{sq}(x)$, e.g. $\text{sq}:x\mapsto \text{sq}(x) := x^{2}$
Other use for this notation is to simply say to what element of $B$ a particular $a\in A$ is mapped to. E.g. $\text{sq}:8\mapsto 64$.

So, what's up with your examples?


*

*Let's begin by your fourth notation $\langle,\rangle: (a,b)\to c$.
It should actually be $$\begin{align}\langle,\rangle : & A\times B \to C \\ &(a,b)\mapsto \langle a,b\rangle \end{align}$$ since that is what the map does. It takes an element $(a,b)$ of $A\times B$ (which is an ordered pair composed by taking an element $a$ of $A$ and an element $b$ of $B$), and then returns an element of $C$ denoted by $\langle a,b\rangle$.

*Of course, for a particular choice of $a\in A$, $b\in B$ and $c\in C$, it could be the case that $\langle a,b\rangle = c$. That is what is meant by your first notation.

*The second notation you bring here is what I explained in the beggining. By $a:b\to c$ you mean $a$ is a map that eats elements of the set $b$ and returns elements of the set $c$. If we restrict ourselves to the convention that sets should be uppercase letters, then it should read $a:B\to C$

*Finally, your third notation $a(b) = c$ is evaluation. It says $a$ is a map from a set $B$ to a set $C$, and that $b$ is an element of $B$, and $c$ is the element of $C$ to which $b$ is mapped under the map $a$. Note this is equivalent to $a:b\mapsto c$.
