Question 207968
Since the general expression is x$y, to find 2$3, simply plug in x=2 and y=3 into {{{(1+x)+y}}} and evaluate the expression. I'll let you do that.



To find out if the operation is closed, you need to ask yourself: if I plug in two arbitrary natural numbers, will I always get a natural number out? The answer is yes. Why? Recall that the sum of two natural numbers is a natural number. Since 1 is a natural number, 1+x is also a natural number. Finally, (1+x)+y is a natural number (given that x and y are natural numbers). So this shows us that $ is closed under the set of natural numbers. 



As for commutativity, we need to determine if x$y=y$x. In terms of the definition, this means that we need to see if {{{(1+x)+y=(1+y)+x}}} (just switch each instance of x and y) is an identity. I'll let you determine that.



To determine associativity, you need to determine if (x$y)$z=x$(y$z). If you can show that (by using the definition), then you can show that $ is associative.