# Binary Operations!

• Oct 7th 2008, 08:24 AM
kbartlett
Binary Operations!
Let S = {a, b, c, d}.
How many binary operations can be defined on S?
I think the answer might be $4^{16}$

How many are commutative?
I think the answer is $4^{10}$

Assume that x = x * a = a * x for all x in the set S.
Then how many binary operations are there?
I think the answer is $4^{9}$

Can any1 tell me if my answers are right or wrong?
• Oct 7th 2008, 09:22 AM
Plato
The answer really depends upon the definitions is use.
How are you defining binary operator?
The most common definition is a function $\phi :X \times X \mapsto X$.
If that is you definition then yes $4^{16}$ is correct for the total number.