#Proving p+r is only divisible by q when q is a factor of r (given q divides p)
neon wagonBOT
glad kindle
This is my proof currently, It’s the first formal proof i’ve written so I would just like to know if it is correct, well written and rigorous.
proven kayak
This is a pretty good first attempt. Importantly, it's enough that you can work on it to make it better.
Your proposition is written as a biconditional. Is that how it's supposed to be? Because you only even tried to prove one of the directions.
Your proposition talks about q/r, but your conclusion talks about r/q.
When you are proving a statement about all p in Z, the first line of your proof needs to be:
Let p be in Z.
(Your first line is also a bit muddied because you have a "for all" symbol, but it's not clear what that's being applied to.)
glad kindle
This is a pretty good first attempt. Importantly, it's enough that you can work ...
thank you
yes i’m using this proposition as part of another proof.
i was trying to say that q|p+r if and only if r/q is in the set of integers
since the form for q|p
is p = kq where k is in the set of integers
and from this i got the form for q|p+r
to p=q(t - r/q), i thought it was ok to say since t is in the set of integers
t - r/q is in the set of integers and therefore r/q must be in the set of integers for the equality to hold
and thus the if and only if
is that sufficient or no?
glad kindle
(Your first line is also a bit muddied because you have a "for all" symbol, but ...
ok thank you, i will correct all these
proven kayak
You're still getting a bit confused between the general and the specific.
You shouldn't be writing general statements like this in the middle of your proof
Proof:
Let p, q, and r be integers.
For the => direction:
Suppose that q|p and q|p+r.
Then ...
(Come to think of it, this isn't true in the opposite direction.)
glad kindle
You shouldn't be writing general statements like this in the middle of your proo...
my prof gave us this statement to use as a definition for q divides p
proven kayak
Right, but you don't need to be stating it like that in the middle of your proof
Also, as written, it's a little unclear whether you're even stating it.
When you say "P => Q", are you claiming P?
You shouldn't be quantifying q here
That's the point of declaring q in the first line.
glad kindle
i see
glad kindle
You shouldn't be writing general statements like this in the middle of your proo...
should i write something like given the definition of…
proven kayak
yeah, you can just put "by definition"
glad kindle
alright
proven kayak
Hold on. I don't think your second last line is true.
4 = 6(2/3) does not mean that 2/3 is an integer
glad kindle
Hold on. I don't think your second last line is true.
oh
glad kindle
4 = 6(2/3) does not mean that 2/3 is an integer
yeah but 4 doesn’t divide 6
this only holds true for q that divides p
glad kindle
do I just write “assume q|p+r is true” or is there a better way of writing that
glad kindle
Better! you should also let r be an integer too.
ok thank you
proven kayak
do I just write “assume q|p+r is true” or is there a better way of writing that
just "assume q|p+r"
proven kayak
just say it while you're making the other assumption
glad kindle
okok
proven kayak
overall, this is much better, and I hope you can see why
you're still missing the step where you actually use that q|p
in order to isolate t-r/q, I'd divide both sides by q
then you can use the assumption to make a claim about the left side
glad kindle
wait why do I have to divide both sides by q?
since we’re given the definition for q|p i was just subbing p+r into p for the definition
proven kayak
Okay, you don't have to. I just thought it was easier. You have to do something.
proven kayak
wait why do I have to divide both sides by q?
since we’re given the definition f...
p+r? We might be talking about different places.
This is still unjustified.
glad kindle
I think so, I found one of my prof's examples
proven kayak
Why did you write this here?
You should make some reference to k near the end
You could argue that the reader can understand the last line, but you could equally argue that the entire thing is obvious and that you shouldn't have to write anything.
The point is to explain something, and the second last line is still unexplained.
It's good that you're persisting though. This is how you learn to write proofs.
glad kindle
Why did you write this here?
im trying to say, something along the lines of
lets assume q|p+r then... (given the definitions and some manipulations) we can prove r/q is in the set of integers
glad kindle
The point is to explain something, and the second last line is still unexplained...
how do i indicate t-r/q represents the k in the definition
glad kindle
It's good that you're persisting though. This is how you learn to write proofs.
thanks