if x,y are elements of the integers and x divides 3, then x,y divide 3?

Question

rmemeber divides not divided by.

Answer 1

A poster is choosing to show off their set theory skills.

The problem is very incomplete and no set theory can make the above problem solvable.

Answer 2

(1) Let X be a set, and let R X × X be a subset of the product
of X with itself. For x, y 2 X, we write x y if (x, y) 2 R
and x y if the pair (x, y) is not in R. We assume that the
following statements are true for all elements x, y and z of X:
(a) x x
(b) x y () y x
(c) ((x y) ^ (y z)) ) x z
For every x 2 X, we define the subset [x] X by
[x] := {y 2 X | x y}.
Using only (a), (b) and (c), prove:
(d) 8x 2 X : x 2 [x]
(e) 8x, y 2 X : (x 2 [y] () y 2 [x])
(f) If x y then [y] [x].
(g) If y 2 [x] then the sets [x] and [y] are equal, i.e.,
8x, y 2 X : (y 2 [x] ) [x] = [y]) .
(h) If x y then the sets [x] and [y] are disjoint, i.e.,
8x, y 2 X : (x y ) [x] \ [y] = {}) .
Once you have proved a statement (d), (e), (f), (g) or (h), you
are allowed to use it in order to prove the others. I think that
it makes sense to prove them in the order stated, but you don’t
have to. If you cannot prove a statement, you are still allowed
to use it for proving the ones below it. Be careful to avoid
circular arguments!

(2) Prove the following statement: If x is an integer such that x2 is
not a multiple of three, then x itself is not a multiple of 3. You
will prove the converse “if x is not a multiple of 3 then x2 is not
a multiple of 3” in Problem 5, and you are allowed to use this
fact for Problem 3.
(3) You can find the proof that p2 is not a rational number in
the book as Example 8.13. It is an application of the proof by
contradiction method. Modify the proof in the book in order
to prove that p3 is not a rational number.
(4) In the arena at the Colosseum there are 100 ravenous lions and
a Christian. The most ravenous lion is offered the opportunity
to eat the Christian. If he declines, then the Christian is
set free, but if he eats the Christian, then that lion is miraculously
converted into a Christian, and the spectacle continues.
Assuming that the rules have been announced to the lions, determine
what happens. (I recommend to try a proof that uses
the principle of induction).
(5) Let Z be the set of all integers, and consider the set
R = {(x, y) 2 Z × Z | x − y is a multiple of 3}.
(a) Show that R satisfies conditions (a), (b) and (c) of Problem
1. Once you have proved this, you know that the
statements (d), (e), (f), (g) and (h) of Problem 1 are also
true for R, and you are allowed to use them.
(b) Any integer n can be written in the form
n = 3k + r,
whith k 2 Z and r 2 {0, 1, 2}, where r is the remainder of
n when divided by 3 (you don’t need to prove this fact).
Prove that for every n 2 Z,
[n] = [0] or [n] = [1] or [n] = [2].
(c) Write down the first four positive elements of each of the
sets [0], [1] and [2].
(d) Prove: if x1 x2 and y1 y2, then
x1 + y1 x2 + y2 and x1 · y1 x2 · y2.
(e) Now prove that if 3 does not divide x then 3 does not divide
x2.
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