# Thread: Basic definition of binary operations

1. ## Basic definition of binary operations

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 $a,b \in A$ and $a \star b \in A$, then $\star$ is a binary operation. How does $f: A \times A \rightarrow A$ mean the same thing? It seems like if A were a set of individual numbers (such as the set $\mathbb{Z}$), 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?

2. ## Re: Basic definition of binary operations

Just a notation $a*b:=f(a,b)$ .

3. ## Re: Basic definition of binary operations

A binary operator is a function of two arguments. Often they are denoted by symbols like +, $\times$, etc. rather than letters and are written using infix notation, e.g., x+y instead of +(x,y).

Originally Posted by beebe
I get the idea that if $a,b \in A$ and $a \star b \in A$, then $\star$ is a binary operation.
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 $\star$ is a binary operator on A means that $\star$ is a function from $A\times A$ to $A$. Declaring the type of $\star$ should come before forming statements like $a\star b\in A$.

Originally Posted by beebe
How does $f: A \times A \rightarrow A$ mean the same thing? It seems like if A were a set of individual numbers (such as the set $\mathbb{Z}$), then f would change A to a set of ordered pairs made from elements of A.
I am not sure if I see what you mean, but if $f:A\times A\to A$ and $x\in A$, then $f(x)$ can be considered as a partially applied function $A\to A$, i.e., a set of ordered pairs. This is used in programming sometimes. However, when $f:A\times A\to A$, the domain of $f$ consists of pairs of elements of A, so, strictly speaking, $f$ has to be applied to the whole pair at once.

4. ## Re: Basic definition of binary operations

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.

5. ## Re: Basic definition of binary operations

Thanks. I misunderstood how the function notation works.