Is Set S Open in R3? A Proof by Using Open Discs

[Cyn]

Homework Statement

I have a set I = {x from R3 : x1<1 v x1>3 v x2<0 x x3>-1}

Homework Equations

Open disc
B (x,r)
(sqrt (x-x0)^2 + (y-y0)^2) < r

The Attempt at a Solution

I have done, for example by x1<1, that let r = 1-x1
Then sqrt ((x-x1)^2 + (y-y1)^2) < sqrt (x-x1)^2) < r = 1-x1
So |x-x1| < 1-x1

x1-1<x-x1<1-x1
2x1 -1 < x < 1

So x<1 satisfy the inequality, so it is open. Is this correct?

 

[andrewkirk]

You seem to have the right general idea, which is to show that the set S={(x1,x2,x3) : x1<1} is open because, any point (x1,x2,x3) in S is contained in the open ball centred on that point with radius (1-x1).

But you have not shown that. All you've done is write a few equations, with no explanations of their relevance, or how they relate to one another, and most of the terms undefined. A proof must tell a story, with a beginning, a middle and an end. And just like how in a novel, the characters must be introduced to the reader, the symbols you use must be explained (defined).