Associativity of Group Operation

[port31]

Homework Statement

Im looking at this example and trying to figure out how they showed it was associative.
They start out with x*y=x+y+1
then they add in z to show it is associative.
x*(y*z)=x*(y+z+1)=x+(y+z+1)+1=x+y+z+2
I don't know how they go from
this x*(y+z+1)=x+(y+z+1)+1
and end up with the equation on the right.
If i solved for x from x*y=x+y+1 and plugged it in I wouldn't get what they got.
Maybe there is some trick that I am missing.

 

[haruspex]

Homework Statement

Im looking at this example and trying to figure out how they showed it was associative.
They start out with x*y=x+y+1
then they add in z to show it is associative.
x*(y*z)=x*(y+z+1)=x+(y+z+1)+1=x+y+z+2
I don't know how they go from this x*(y+z+1)=x+(y+z+1)+1
and end up with the equation on the right.

They're simply applying the rule for *. If the general rule is x'*y'=x'+y'+1, and you have x*(y+z+1) then you can match up x' with x and y' with (y+z+1):
x*(y+z+1) = x'*y' = x'+y'+1 = x+(y+z+1)+1

If i solved for x from x*y=x+y+1

What do you mean you 'solved it'? The * doesn't stand for normal multiplication here. x*y=x+y+1 is defining an operation '*'.

 

[port31]

oh ok i see now, thanks for your help.