Formal definition of a mapping confusion This is rather basic introductory to algebra stuff but I don't seem to understand the formal definition of a mapping, here is what it says in my notes:
by a mapping/function $f: A \to B$ we mean a subset $f \subset A \times B$ satisfying the following conditions:


*

*$\forall a \in A$, $\exists b \in B$, s.t. $(a,b) \in f$

*$(a,b) \in f$ and $(a,b') \in f \Rightarrow b = b'$


I have a few problems with this:
firstly, I know the definition of $A\times B$ (well, I know what it is) but what does it really mean? I understand all the other set operations, just not this one. My book defined it as this $A\times B = \{(x,y) | x\in A \wedge y\in B \} = \{(x,y) | P_A(x) \wedge P_B(y) \}$ but I don't particaly know what it means, it looks very similar to $A \wedge B$.
Secondly, what does it mean by a subset of f in $A\times B$?
Thirdly, is $(a,b)$ an ordered set? Not really sure about this notation either.
If anyone could clear this up - thank you
 A: $(a,b)$ is an ordered pair of elements $a$, $b$, or, a sequence of length $2$, if you prefer. Usually in ZFC set theory $(a,b)$ is defined as the set $\{\{a\},\{a,b\}\}$ (which exists by the pair axiom), which has the most important property of ordered pairs that
$$(a,b)=(c,d) \iff (a=c\,\land\, b=d)$$
Then, $A\times B$ is the set of all ordered pairs of the form $(a,b)$ with $a\in A$ and $b\in B$. 
 (For finite sets, we have $|A\times B|=|A|\cdot|B|$ where $|S|$ denotes the number of elements of set $S$.)
A subset $R$ of $A\times B$ is also called a (binary) relation between sets $A$ and $B$, and it contains some pairs $(a,b)$. For example, the equality relation ('$=$') on a set $A$ can be considered as the diagonal relation $\{(a,a)\mid a\in A\}\,\subseteq A\times A$.
A relation $F\subseteq A\times B$ is called a function, if for each $a\in A$ there is exactly one $b\in B$ such that $(a,b)\in F$. When $F$ is regarded as a mapping, the unique $b$ which satisfies $(a,b)\in F$ is considered the image of $a$ under $F$. 
(The usual notations like $F:\Bbb R\to\Bbb R,\ F(a):=a^2$ or $F:a\mapsto a^2$ can be translated into the language of set theory as $F=\{(a,a^2)\mid a\in \Bbb R\}$.)
A: Lets use a concrete example to try to understand the definition. Just take A and B to be the real numbers, for now. Then $A \times B$ is the set of all ordered pairs of real numbers, so it's just a plane. You can think of the set $f \subset A \times B$ as the graph of the function $f$. 
Property #1 says that every point $a \in A$ has some point of the graph that is above it.
Property #2 says that if there are two points above some $a \in A$, then these two points must be the same. 
So, in summary, #1 and #2 say that every $a \in A$ has exactly one point of the graph that is above it. The vertical height of this point is what we typically denote by "$f(a)$".
This is just an example, so you can't use it to prove any general theorems, but it might help you understand the concepts. 
