1. ## Binary operation

Determine whether the following is a binary operation:

S={1,-2,3,2,-4} a*b=|b|

(where a*b is the "generic" notation for any binary operation)

Now to my understanding a binary operation takes two elements from a set, and creates a third from the same set via a binary operation. Now |b|
requires the first element to be b and the second elemet to also be b.
But the magnitude is b times b and then taking the square root, so is this classified as more than 1 binary operation.

Also for the difinition of a binary operation does the third element have to be unique? or is this only for a one to one map?

2. Originally Posted by ulysses123
Determine whether the following is a binary operation:

S={1,-2,3,2,-4} a*b=|b|

(where a*b is the "generic" notation for any binary operation)

Now to my understanding a binary operation takes two elements from a set, and creates a third from the same set via a binary operation. Now |b|
requires the first element to be b and the second elemet to also be b.
But the magnitude is b times b and then taking the square root, so is this classified as more than 1 binary operation.

Also for the difinition of a binary operation does the third element have to be unique? or is this only for a one to one map?
If we take S to be the integers then this is clearly a binary operation. A binary operation takes an ordered pair and does something to this pair to get a single element out,

$\displaystyle S \times S \rightarrow S$
$\displaystyle (a, b) \mapsto c$.

There is nothing to say that this element c must depend on both a and b, but it must be in S. For instance, $\displaystyle a.0=0$ (where is the usual multiplication).

However, you are not in the integers. You are in a given set, S. This is not a binary operation. To see this, try multiplying some of your elements together. Remember, their product must be in S...

Now, you ask whether |b| is a binary operation? It isn't, as it takes an element from the set-a single element-and squares it, then finds its square root. It has an input of one element and an output of one element, so it is what is called a unitary operation.

And yes, the third element must be unique. It would not be a well-defined operation otherwise...