Question

Let x and y be distinct real numbers. Prove there is a neighborhood P of x and a neighborhood Q of y such that P intersection Q = emptyset

Answer 1

What I got so far...

Suppose we have a neighborhood P of x
\[(x- \epsilon, x+ \epsilon) \subseteq P\]
and a neighborhood Q of y
\[(y- \epsilon, y+ \epsilon) \subseteq Q\]

Answer 2

so \[P \cap Q = \emptyset \] either means that set P and set Q have nothing in common.
or..
both sets are empty meaning no elements in set P and no elements in Set Q

Answer 3

Prove that water is wet and the sky is blue.

Answer 4

take something less than half the distance between them

Answer 5

wow so many people in here. I should post more often. I'm usually at stackexchange

Answer 6

Is epsilon supposed to be repeated across x and y?

Answer 7

Take r = |x-y|/2 (or any smaller positive number), and consider the neighborhoods P = {t in R : |t - x| < d} and Q = {t in R : |t - y| < d}.

Answer 8

`Let x and y be distinct real numbers. Prove there is a neighborhood P of x and a neighborhood Q of y such that P intersection Q = emptyset `

basically they want you to show that no matter how close x and y are, there's "empty space" between them (or the two neighborhoods)

this isn't a formal proof but a way to think of it

Answer 9

so.. if x and y are close, there is going to be a gap .
so I have to take the distance between x and y. ???

Answer 10

@inkyvoyd

Answer 11

let's say x < y

Answer 12

and for some epsilon, we can form these intervals

Answer 13

as long as x+e < y-e then you will have empty space

this is definitely possible if e is small enough

Answer 14

ah I remember those intervals.
and then there's is going to be an empty space, but e needs to be very small.

Answer 15

idk how to make it more rigorous but others on here might have an idea

Answer 16

well we need to take less than half the distance... also we are dealing with x and y being distinct real numbers. I'm wondering what the definition that abbot had was called because that's the missing piece I need. The book I have only did what I've type
which is

A set Q of real numbers is a neighborhood of a real number x
iff Q contains an interval of positive length centered at x-that is, iff there is
e > 0 such that (x-e,x+e) C Q.

yeah I just copy pasta-ed the definition so no latex in there

Answer 17

take something less than half the distance between them

Answer 18

oh very true, as long as e < (y-x)/2 then it works

Answer 19

where did
consider the neighborhoods "P = {t in R : |t - x| < d} and Q = {t in R : |t - y| < d}. " come from? I can't find that version in my book.. just the intervals contained in a set Q and the neighborhoods definition

ok so do I take the less than half the distance between them....between the x+e and the y-e?

Answer 20

no, you're taking e to be LESS THAN HALF the distance

Answer 21

you can pick any epsilon you want... just have to show that with the epsilon you pick, the intervals don't intersect

Answer 22

by intervals i mean [x-e,x+e] and [y-e,y+e]
technically speaking epsilon doesn't even have to be the same across x and y, but why make life hard

Answer 23

so the epsilons can be different . . . there could be a bunch of epsilons but only one of the epsilons will result in not having the intervals intersect.

Answer 24

let's say those eclipses are centered, and are the same size.