Trouble showing the addition of the fractional parts are a group.

Question

I'm working on some released exams, in particular this one and I am stuck on 4B.

I know I need to prove three properties to show that it is a group:

1. The operation is associativity
2. There exists an identity
3. There exists an inverse.

I can find a solution for 2 and 3 somewhat easily:

For 2:

Let $x$ be an element belonging to the equivalence class. Then $[x + 0] = [x]$, which implies $[0]$ (not actually zero itself, but zero's equivalence class) is the identity element.

For 3:

Let $[x]$ be an element belonging to the equivalence class. Then $[x + (-x)] = [0]$ which implies $[-x]$ is the inverse of [x].

I am having trouble with the last part, associativity. I'm virtually lost:

Let $x,y$ and $z$ be elements of arbitrary equivalence classes. Then:

$$[[x + y] + z] = ?$$

I don't know what to do from here. Any ideas? Also, when the question states:

Be sure to check that the operation is well defined.

I don't understand what I am supposed to be looking for. My first though is that there isn't a unique identity or inverse... But the more I stare at the problem, I believe there is.

Answer 1

Note that in your answer, $$[[x+y]+z]$$ does not make sense. To work it out you would have to begin with $$[x+y]+z\ ,$$ which is a set plus an individual number, which cannot be done. The associative law states that $$(\,[x]+[y]\,)+[z]=[x]+(\,[y]+[z]\,)\ :$$ if you carefully use the definition of addition, and make sure you do the steps in the right order, you should find this is not too hard to prove.

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Re: well-defined, let's start with an example of something which is not well defined. Suppose you try to order your equivalence classes by defining $$[x]>[y]\quad\hbox{if and only if}\quad x>y\ .$$ Then we have, for example, $$[0.5]>[0.2]\quad\hbox{and}\quad [0.5]<[1.2]\ .$$ But this does not make sense, because the equivalence class $[1.2]$ is the same as $[0.2]$, so we have said that it is both less than and greater than $[0.5]$. We say that the order specified here is not well-defined: the point is that since $[0.2]=[1.2]$, anything sensible that we say about this class cannot depend on the specific numbers $0.2$ or $1.2$.

Looking at addition, we have for example $$[0.2]+[0.5]=[0.7]\ .$$ If we calculate $[1.2]+[0.5]$ it is really the same sum, so we should get the same answer. And indeed, $$[1.2]+[0.5]=[1.7]=[0.7]\ .$$ What you have to prove is that addition always works "properly" in this respect, that is, $$\hbox{if}\quad [x_1]=[x_2]\ \hbox{and}\ [y_1]=[y_2]\ ,\quad\hbox{then}\quad [x_1+y_1]=[x_2+y_2]\ .$$ Good luck!