Is this operation associative?

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Question from patrick, a student:

Associative test: Can you explain the following to me?

Is the following operation associative?: x*y=x+y+1
1) x*(y*z)=x*(y+z+1)=x+(y+z+1)+1=x+y+z+2
2) (x*y)*z=(x+y+1)*z=(x+y+1)+z+1=x+y+z+2

The answer is yes as 1) = 2)

My specific questions are:
1) How x*(y*z)=x*(y+z+1)=x+(y+z+1)+1 ?
2) How (x+y+1)*z=(x+y+1)+z+1?

Thank you!!

Hi Patrick,

I don't know what $x, y$ and $z$ are but I am going to assume they are numbers. You have a new operation, asterisk, that works as follows. To "asterisk" two numbers you add the two numbers and then add 1.

Look at 1) $x*(y*z).$ You do what is inside the parentheses first, $y*z,$ which, according to the "asterisk" rule is $y + z + 1.$ So now you have $x*(y + z + 1).$ According to the "asterisk" rule you are to add $x$ to $(y + z + 1)$ and then add $1.$ Thus $x*(y + z + 1) = x+(y + z + 1) + 1.$

I hope this helps,
Penny

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