On the set N of non-negative integers This is the fill in blanks type question asked in one exam
On the set N of non-negative integers, the binary operation ______ is associative and non-commutative.
i am not getting any standard operator.Is there any such operator exists or question is wrong?
 A: If the OP is not limited to using "common" arithmetic operations, then projection gives an easy example. Namely, let $$x*y=x.$$ Then this is associative $$(x*y)*z=x*z=x\quad\mbox{ and }\quad x*(y*z)=x*y=x$$ but not commutative $$1*2=1\quad\mbox{ but }\quad 2*1=2.$$

If we're restricted to using "common" operations, then it's unclear whether the question can be answered - what exactly is a "common" operation? However if we drop the associativity requirement, then exponentiation is a good example: $2^3=8\not=9=3^2$. Of course exponentiation is not associative, though ($(2^2)^3=4^3=64$ but $2^{2^3}=2^8=256$) so (unless the problem's copied incorrectly) this doesn't work.
A: None of the standard operators has those properties. However it is not hard to make up an operator with those properties. The simplest possibility is what in the C programming language is known as the comma operator: It just gives the second argument, $(a,b)\mapsto b$. I'll instead write it as $*$ here.
The operator is clearly associative:
$$(a*b)*c = c = a*(b*c)$$
But is is not commutative: For $a\ne b$ we have
$$a*b = b \ne a = b*a$$
