A binary operation on A is a rule that assigns to every pair
of elements of A a unique element of A. We are used to addition
and multiplication of real numbers and these are examples of binary operations.
Formally, a binary operation is a function 
.
For example, addition on the integers could be defined as the function
with our familiar way of
producing the answer. Thus +(2,3) = 5 and +(-4,11) = 7. This looks a
little too weird and we are probably happier to see 2+3=5 and -4 + 11 =5.
Likewise, if * is a binary operation on a set then we will write a*b rather
than *(a,b).
One problem that we have is how shall we name these binary operations? When we
talk about the binary operations "multiplication" and "addition" we usually
think about the operations on the real numbers. Rather than invent new words
for every new binary operation we tend to use "multiplication" or "addition"
but realize that we may not mean our usual operation. As an example, we can
define a multiplication * on the positive integers by 
. Then as
3*2 = 9 we can speak of "3 times 2 is 9" (or had we named this operation
addition "3 added to 2 is 9". The context of the problem should make it clear.
If of course we happen to be discussing three binary operations then it would
make sense to invent new names to reference each operation.
The idea of a binary operation is just a way to produce an element of a set
from a given pair of elements of the same set. In the case of a finite set
we could list the rule in a table which we'll call a multiplication
table.
(For an infinite but countable set we might be able to imply a multiplication
table.
)
Here is an example of a binary operation (called !)
on 
.
If a set has n elements then there are 
binary operations that
can be defined on that set. For our two element set the operation ! is one
of 16 binary operations possible.
To see this, a multiplication table has
n rows and n columns so that there are 
entries to be filled.
There are n ways to fill each entry and so the total number is found by
multiplying n by itself times.
In studying binary operations on sets, we tend to be interested in those operations that have certain properties and we discuss this further in the next section.