@rahul
subtraction will voilate closer property, it is said that "on a set of non negative integers"...but subtraction could lead to negative integers.....
for eg: 2*4 = 2-4 = -2// not a non negative integer.
@rahul+sharma+5, subtraction by default is not associative either.
The most important associative operation that's not commutative is function composition. If you have two functions f and g, their composition, usually denoted f∘g, is defined by (f∘g)(x)=f(g(x)). It is associative, (f∘g)∘h=f∘(g∘h), but it's usually not commutative. f∘g is usually not equal to g∘f.
The most important associative operation that's not commutative is function composition.
If you have two functions f and g, their composition, usually denoted f∘g, is defined by
(f∘g)(x)=f(g(x)). It is associative, (f∘g)∘h=f∘(g∘h),
but it's usually not commutative. f∘g is usually not equal to g∘f.
For our case suppose $\forall$x $\in$ N of non-negative integers, if f(x)=x^{2} and g(x)=x+1, then (f∘g)(x)=(x+1)^{2} while (g∘f)(x)=x^{2}+1, and they're different functions.
nice example (y) If $f∘g$ exists then $g∘f$ might not even exists.
The tests are there but it ain't free. Cost is...