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

Re: Basic definition of binary operations

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

Re: Basic definition of binary operations

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

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Originally Posted by

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

Quote:

Originally Posted by

**beebe** How does $\displaystyle f: A \times A \rightarrow A$ mean the same thing? It seems like if A were a set of individual numbers (such as the set $\displaystyle \mathbb{Z}$), 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 $\displaystyle f:A\times A\to A$ and $\displaystyle x\in A$, then $\displaystyle f(x)$ can be considered as a partially applied function $\displaystyle A\to A$, i.e., a set of ordered pairs. This is used in programming sometimes. However, when $\displaystyle f:A\times A\to A$, the domain of $\displaystyle f$ consists of pairs of elements of A, so, strictly speaking, $\displaystyle f$ has to be applied to the whole pair at once.

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.

Re: Basic definition of binary operations

Thanks. I misunderstood how the function notation works.