Mathematical operation with unique result Does a mathematical operation 'op' exist so that there is a unique result for every expression? so:
a op b ≠ b op a
a op b ≠ c op a
a op b ≠ c op d
a op b = a op b
Thanks in advance.
 A: The Cantor pairing function $(a+b)(a+b+1)/2+b$ is an example on $\mathbb{N}$
A: Yes, there are such operations.  Apart from trivial examples, we need an infinite set.  For instance, work with the set $\mathbb{N}$ of natural numbers, and define the operation $\ast$ by  $a \ast b=2^a3^b$ for all natural numbers $a$, $b$. 
There are more interesting examples, with additional properties.  It is useful from now on to work with $\mathbb{N}_0$, the set of non-negative integers.
On $\mathbb{N}_0$, define the operation $\ast$ by $a\ast b=2^a(2b+1)-1$. Then it turns out not only that $a\ast b \ne c\ast d$ if $(a,b)$ and $(c,d)$ are distinct ordered pairs, but that in addition the operation $\ast$ maps $\mathbb{N}_0\times \mathbb{N}_0$ onto $\mathbb{N}_0$.  That means that for every $c \in \mathbb{N}_0$, there is a unique ordered pair $(a,b)$ such that $a\ast b=c$.  Verification of this fact is easy.  Given any non-negative integer $c$, express $c+1$ as a power of $2$ times an odd number. That uniquely determines $a$ and $b$.
The operation $\ast$ on $\mathbb{N}_0$
defined by 
$$a\ast b=\frac{(a+b)(a+b+1)}{2}+a$$
also maps $\mathbb{N}_0\times \mathbb{N}_0$ onto $\mathbb{N}_0$, and has the property you seek.  These various pairing functions are even useful.  You may want to look at this link.  This mapping $\ast$ has the nice property of being a polynomial mapping.  Verification that it has the required properties takes more effort than in the preceding example.
We can also find operations of the type you seek on the reals, indeed on any infinite set, but the details get more complicated.
A: To give an example on the reals, or at any rate on the reals $x$ such that $0\le x\lt1$, consider $$0.a_1a_2a_3\dots{\rm\quad op\quad }0.b_1b_2b_3\dots=0.a_1b_1a_2b_2a_3b_3\dots$$ where the $a_i$ and $b_i$ are the decimal digits of the two numbers. 
A: Your operation is essentially the operation of forming ordered pairs.
