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What is (mod n) in: a congruent b (mod n)

https://en.wikipedia.org/wiki/Equivalence_class
I got this bit. Here's a rough explanation.
1. You have a set S and a bunch of elements belonging to S
2. element1 == element2 can be written as element1 is congruent to element2
3. We do this by creating Equivalence Classes (which are small subsets) based upon some rule: 'color of the car'
4. Then we stick the cars with the same color in an the subset/equivalence class.

What I DON'T get is this bit:

a congruent b (mod n)

basically this means 'a' is the same as 'b' - great, what's the rule?

(mod n)? That's not a rule like.. 'all cars are green'? So my question is, what is 'mod n'

Reply 1

veekm

okay never mind I got it after asking on IRC :tongue:

Basically (mod n) means 'they share the same reminder'
Therefore, a congruent b (mod n) is code for :
a = pn + r
b = qn + r
(above two lines are division by 'n' in sekret-code)

SO basically you are rather impolitely saying, divide 'a' and 'b' by n, and if the remainder is the same, then they are equivalent.

Reply 2

veekm

Now I have another query:
How do I correlate (intuitively) equivalency classes with modular arithmetic with division and apply all this to the clock in my earlier Q. With cars, the rule is two cars are the same if the color is green. With integer congruence, 'a' and 'b' is the same if the remainder is the same when 'divided' by 'n'.. but that's still in the realm of numbers.. how do i get a car or some color in there.. i'll try with object..

https://en.wikipedia.org/wiki/Modular_arithmetic

implies

then we do

and cause confusion

so, i've always (as of 1 hr) imagined division as a mechanism for building an object.
------------------------
take the above line segment - when you write 9 = 4.2 + 1 (maths calls this division of 9 by 2 or 4) BUT MORE importantly..
we are building a line (above) by using a unit line. So.. there's this tiny unit/sample line and I double it, then quadruple the result to get close to my complete target line, then I add one unit to complete (note they are integers because I can pick and choose how big my unit line is to satisfy integer multiplication).

I have an equivalency between two objects 'a' and 'b' if object 'r' is the same, but object 'n' must also be the same.
Object 'p' 'q' and the unit can be different - but surely i get TWO unit objects (one for each equation)?

I'm totally confused about picking the unit objects.. anyone here who has undergone this can clarify perhaps how you understood all this in harmony..

Reply 3

veekm

basically what i'm asking is - when you see this:

how do you handle 1? 'r' is r.1, a is a.1
1 is our unit of measure carefully picked to satisfy that equation. If 'a' were an actual object like a table or a line.. 1 would imply some smaller fraction of the bigger object picked so that a, p, n, and r could exist as integers. But when there's another equation like

that's a different real world object WHICH obviously means we have a different sort of 1?
(i can't sk this on IRC boo hoo.. i'll try usenet eventually.. could someone read all that crap and genius it for me)