Question 207968
An operation $ has been defined over the set of natural numbers. in each case, find 2$3, determine whether or not the set of natural numbers is closed under $, determine if $ is commutative and determine if is associative.
---------------------------- 
x$y=(1+x)+y
Then 2$3 = (1+2)+3 = 6
------------------------------------
Naturals closed under $ ?
Let a and b be natural numbers
Then a$b = (1+a)+b = 1+(a+b)
Since the natural numbers are closed under addition,
1 + (a+b) is is a natural number. So the set of 
natural numbers is closed under $.
------------------------------------
Is # communtative?
Show that a$b = b$a
(1+a)+b = (1+b)+a
1 + a + b = 1+ b + a
Since the naturals are commutative,
1 + a + b = 1 + a+b
--------------------------
Is $ associative?
Show that (a$b)$c = a$(b$c)
============
Can you do that?
Cheers,
Stan H.