What does $f: A \times A \to A$ mean? What does $f: A \times A \to A$ mean?
Can you give some examples please?
 A: Normally this means that $f$ is a function with domain $A\times A$ and codomain $A$.  One way to represent a function is with ordered pairs.  For this example, it would be a set of ordered pairs $((a,b),c)$, where $(a,b)\in A\times A$ and $c\in A$.  Formally, $$f\subseteq (A\times A)\times A$$
To be a function, each element of $A\times A$ must be represented exactly once as the first element of the ordered pairs contained in $f$.
A concrete example would be for $A=\mathbb{R}$ and $f:(x,y)\rightarrow x+y$.
A: It means a function taking two arguments from the set A (formally: A pair of elements of $A$; $A\times B$ is the set of pairs $(a,b)$ where $a\in A$ and $b\in B$; here $B=A$) and results in another element of $A$.
Example: If $A = \{1, 2\}$, then $f$ might be the function which gives the smaller of the two arguments, that is $f(1,1) = f(1,2) = f(2,1) = 1$ and $f(2,2) = 2$.
Another example: $+:\mathbb R\times\mathbb R\to\mathbb R$ is a function (whose application is written is the special way $a+b$ instead of $+(a,b)$) which takes two real numbers and gives their sum.
A: This is the way we thing of a binary operation like addition or multiplication of numbers, or of matrices, or polynomials. The function takes an ordered pair of numbers or objects and combines them to give a single number or object of the same kind.
We could, instead, have ideas like subtraction or division or the highest common factor of two integers or the vector (cross) product.
When we make things more formal we need to be clear which objects we have in view - integers, real numbers, polynomials with real coefficients, $2x2$ matrices with rational entries etc. It is that clarity which is expressed by choosing the symbol $A$ to represent the kind of object we are dealing with.
You can perhaps see how important the construction is - and indeed it enables us to generalise operations like addition and multiplication. You will also come across functions which go like $$f: k \times V\to V$$ which express the fact that we can do things like multiply a vector by a scalar and get another vector.
Sometimes having these basic examples in your head can be a great help in decoding the language. Other times you will have to be careful, because the examples are quite special and have properties which don't always apply in the general case.
