Proving a-b=9 for natural numbers in a set of 100 with a card of 55?

[Andrax]

Homework Statement

let E be set of natural numbers where 0<n≤100 Card E = 55
Prove that there exists atleast 2 numbers a and b in each set where a-b=9

Homework Equations

$\existsk1$$\in$[[ ]] x1=9k1+r1 r1$\in$[[ ]]
$\existsk2$$\in$[[ ]] x1=9k2+r2 r2$\in$[[ ]]
.
.
.
$\existsk55$$\in$[[ ]] x1=9k55+r55 r2$\in$[[ ]]
(using derkil theory usually solves these kinds of problems )
but the above didn't help me, anyway i need to make sure that i could get any numbers from 1 to 100 in every equationi tried doing this
$\existsk1$$\in$[[1, 9 ]] x1=9k1+r1 r1$\in$[[1,54 ]]
$\existsk2$$\in$[[1 , 9 ]] x1=9k2+r2 r2$\in$[[ 1,54 ]]
.
.
.
$\existsk55$$\in$[[1 ,9 ]] x1=9k55+r55 r2$\in$[[1,54 ]]
we have 55 r and these r are all in a setB Card b = 54
then atleast there exists ri and rj where ri = rj
so there exists xi and xj such as xi - xj = 9(ki-kj)+ri-ri
xi-xj=9(ki-kj)
now this sadly doesn't get me anywhere because ki - kj must be equal to 1 or -1 .. ( i hope you guys are getting what I'm trying to do here)
anyway i randomly noticed that 54/6 = 9 so i tried another way ..
$\existsk1$$\in$[[ [2] , [3] ]] x1=54/k1+r1 r1$\in$[[ ]]
$\existsk2$$\in$[[ [2] , [3] ]] x1=54/k2+r2 r2$\in$[[ ]]
.
.
.
$\existsk55$$\in$[[ [2] , [3] ]] x1=54/k55+r55 r2$\in$[[ ]]
now tis way "might" work if and only if the ki and kj are different we will have
xi-xj= 54/ki - 54/kj
=(54ki-54kj)/kikj
but the problem on this one is that i can't find a set which contains r's and it's card <55..
i really need help on this problem

 

[mfb]

I don't understand your notation.

You could consider the sets {1,10,...}, {2,11,...}, ...
Can you calculate the maximal cardinality of a set where no two numbers have a difference of 9?

 

[Andrax]

Thanks I knew I was just fooling IA used the same notation in a similar exercise

 

[Mark44]

Homework Statement

let E be set of natural numbers where 0<n≤100 Card E = 55
Prove that there exists atleast 2 numbers a and b in each set where a-b=9

Homework Equations

$\existsk1$$\in$[[ ]] x1=9k1+r1 r1$\in$[[ ]]
$\existsk2$$\in$[[ ]] x1=9k2+r2 r2$\in$[[ ]]
.
.
.
$\existsk55$$\in$[[ ]] x1=9k55+r55 r2$\in$[[ ]]
(using derkil theory usually solves these kinds of problems )
but the above didn't help me, anyway i need to make sure that i could get any numbers from 1 to 100 in every equationi tried doing this
$\existsk1$$\in$[[1, 9 ]] x1=9k1+r1 r1$\in$[[1,54 ]]
$\existsk2$$\in$[[1 , 9 ]] x1=9k2+r2 r2$\in$[[ 1,54 ]]
.
.
.
$\existsk55$$\in$[[1 ,9 ]] x1=9k55+r55 r2$\in$[[1,54 ]]
we have 55 r and these r are all in a setB Card b = 54
then atleast there exists ri and rj where ri = rj
so there exists xi and xj such as xi - xj = 9(ki-kj)+ri-ri
xi-xj=9(ki-kj)
now this sadly doesn't get me anywhere because ki - kj must be equal to 1 or -1 .. ( i hope you guys are getting what I'm trying to do here)
anyway i randomly noticed that 54/6 = 9 so i tried another way ..
$\existsk1$$\in$[[ [2] , [3] ]] x1=54/k1+r1 r1$\in$[[ ]]
$\existsk2$$\in$[[ [2] , [3] ]] x1=54/k2+r2 r2$\in$[[ ]]
.
.
.
$\existsk55$$\in$[[ [2] , [3] ]] x1=54/k55+r55 r2$\in$[[ ]]
now tis way "might" work if and only if the ki and kj are different we will have
xi-xj= 54/ki - 54/kj
=(54ki-54kj)/kikj
but the problem on this one is that i can't find a set which contains r's and it's card <55..
i really need help on this problem

As posted, this is pretty much gibberish.

Thanks I knew I was just fooling IA used the same notation in a similar exercise

Were you asking a serious question? It's to tell from the work you show.