Question

Why is this relation not a total order?
A = ℝ, R = {(x,y)∈ℝxℝ | (|x| < |y| or x = y) }

Answer 1

I need to find a particular x and y so that (x,y)∉R and (y,x)∉R

Answer 2

the relation is already a partial order btw

Answer 3

Obviously I can't choose x and y to be the same because that would satisfies the condition x = y.

So either x < y , or x > y. but it seems either (x,y) or (y,x) will be in R in both cases.

Answer 4

how about opposites

Answer 5

O.O ahaha how could I not think of that!!??

Answer 6

think about it
if (x,y) is in R then |x|<|y|
if (y,x) is in R then |y|<|x|
this impllies |y|<|x|<|y|
which implies |x|=|y|
which is true for two certain cases
can you guess which cases

Answer 7

you just saved me from going insane :DD thank you

Answer 8

np

Answer 9

btw, how does |y|<|x|<|y| implies |x| = |y| ?

Answer 10

how can something be less than something else and also greater than it?

Answer 11

oh...

Answer 12

huhmm... but |y|<|x|<|y| is just false. Why would it imply anything?

Answer 13

it is false but i i said that part imply the other part because your relation also contains that one equality part about x=y

Answer 14

|x|=|y|
implies x=y or x=-y

x=-y would be the one I want to look at because that is the equality my relation did not have

Answer 15

x=-y means I want x and y to be opposites

Answer 16

ah i see

Answer 17

now this |y|<=|x|<=|y|
isn't false because it is true only when |x|=|y|

Answer 18

if you wanted to replace what i said with that i wouldn't see a problem with it

Answer 19

since x=y implies |x|=|y|

Answer 20

I see. Thanks again

Answer 21

np