Partition Math Help: Understanding x~y on Z

[chocolatelover]

Homework Statement

Describe the partition associated with the following:

On Z, we define x~y if and only if x-y is divisible by 3

Homework Equations

The Attempt at a Solution

Could someone please give me a hint? I don't understand what I'm supposed to do.

Thank you

 

[dynamicsolo]

You're supposed to break up the set of integers (Z) into distinct sets. Pick an integer and consider what other integers would have a difference from the one you chose that is evenly divisible by 3. What integers aren't in this set? Pick another integer among those not in your first set; what integers would have a difference from that one that is evenly divisible by 3? Are there any integers left? What happens with those?

At what point in this process have you taken care of all the integers? What do your sets look like? If the sets are distinct and not overlapping, you have a partition!

(Hint?: there is another name and way of describing what you are doing with this partition, but I don't know if you've talked about it in your courses so far...)

 

[chocolatelover]

Thank you very much

Would this work? {{3}, {6}, {9}, {12}, {15}, {3n+3}}

Thank you

 

[dynamicsolo]

Well, {3, 6, 9, 12, 15, ... 3n} would certainly be one of the sets in the partition. (Don't forget that zero and negative integers are involved as well.) What would the others be?

Are you familiar with modular arithmetic? (I ask because it doesn't turn up in some curricula.) If you are, there's a succinct way to describe what you need to do.

 

[chocolatelover]

Thank you

So, it would be 3n and not 3n+1?

Thank you

 

[dynamicsolo]

Alright, you have one set which can be described as {3n}, which is

{..., -9, -6, -3, 0, 3, 6, 9, ...} .

What would the set {3n+1} be?