Just a notation .
Sorry this is such a basic question, but I'm studying all on my lonesome.
I'm struggling with the function/mapping definition of binary operations. I get the idea that if and , then is a binary operation. How does mean the same thing? It seems like if A were a set of individual numbers (such as the set ), then f would change A to a set of ordered pairs made from elements of A. Also, is f the binary operator in the definition?
A binary operator is a function of two arguments. Often they are denoted by symbols like +, , etc. rather than letters and are written using infix notation, e.g., x+y instead of +(x,y).
Before a mathematical object can be used, it has to be declared, i.e., it should be given a definition or at least a type: an integer, a real, a function, a set, etc. Saying that is a binary operator on A means that is a function from to . Declaring the type of should come before forming statements like .
I am not sure if I see what you mean, but if and , then can be considered as a partially applied function , i.e., a set of ordered pairs. This is used in programming sometimes. However, when , the domain of consists of pairs of elements of A, so, strictly speaking, has to be applied to the whole pair at once.
in some cases (particularly when A is finite), the function is f is listed explicitly, with rows labelled "a" and columns labelled "b". this gives a matrix with a*b in the i,j-th postion. such a matrix is called a multiplication table, and may be familiar to you from grade-school.