On Tuesday, February 13, 2018 at 11:02:51 AM UTC-5,
On Monday, February 12, 2018 at 11:03:49 PM UTC-5,
On Monday, February 12, 2018 at 8:18:11 PM UTC-5,
Post by Simon RobertsI don't even know the fundamentals.
If
A || C
and
B \| C
does THIS imply
(A-B) \| C ?
-- where A || C means "A divides C" and B \| C means
"B does not divide C" --
The answer to your question is "Yes", those two do
imply the last.
I think the easiest way to see this is to use modular
arithmetic, which you refer to later.
If A divides C, then C mod A = 0
If B doesn't divide C, then C mod B = k
for some k != 0
(A-B) mod C
= ((A mod C) - (B mod C)) mod C
= ( 0 - k ) mod C
!= 0 , because k != 0
Therefore, (A-B) doesn't divide C
If we want to prove this more formally, we would want
formal definitions of "divides" "mod" and so on.
Off-hand, I don't know if there are common definitions,
but we know what we want them to mean.
"x divides y evenly"
x | y :<->
Ez:( x*z = y )
"z is the remainder of y divided by x"
y mod x = z :<->
( x*z =< y ) &
Aw:(w > z) -> ( x*w > y )
Lemmas
0 =< z < x
x | y -> y mod x = 0
y mod x = 0 -> x | y
Let x | y = z and z != 0. Then
(-x) mod y = (x - z)
Let x | y = z and x' | = z'. Then
(x + x') mod y = (z + z') mod y
What is the standard notation (in text) for
"divides (evenly)" and "does not divide (evenly)"
before I start explaining mine?
What I've seen is a vertical bar | with the divisor
on the left (as I wrote above). There could be other
notations. If you just say what you mean, that would be
enough -- though a good rule of thumb is to follow standard
notation, because of its familiarity.
All the discussion of notation is not a problem,
but all I wanted was to know what || and \| mean
when you use them here.