Question

Is this a linear order on N ?
T, where m T n iff m < 2n

Answer 1

Well to start off I am not sure what linearly ordered means?
Does it mean that \( x \neq y\)?

Answer 2

like its antisymmetric and \( x \neq y\) ?

Answer 3

Linear order is just another word for a total order. So we need antisymmetry, transitivity, and totality.

Answer 4

what is totality?

Answer 5

mTn or nTm but not both.

Answer 6

ohhhhhhhhh so its not comparable okkkkk

Answer 7

Actually, it could be both, I just got confused.

Basically, if you choose any two elements of your set, they must be ordered. Speaking of which, is this defined over the reals?

Answer 8

sorrryy left that out. Its defined over N

Answer 9

First up, we need antisymmetry.

Suppose \(n=4\), and \(m=5\). Clearly \(4<10\) and \(5<8\), but \(4\neq 5\). Hence this is not antisymmetric, and is not a linear/total order.

Answer 10

Like can u explain what u just did.

Answer 11

I am not sure what linear order is i guess

Answer 12

For it to be a linear order, we need it to be antisymmetric. That means we need \[mTn\;\;\text{and}\;\;nTm \implies n=m\]What I showed above was a contradiction to this. I chose \(n,m\) such that \(n \neq m\), and \(mTn\) and \(nTm\). Hence, it can't be antisymmetric. Since it's not antisymmetric, it can't be a linear order.

Answer 13

ohhhhhhhhhhh i sssssseeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

Answer 14

wait i have a question then

Answer 15

So wld u say that this is antisymmetric
R xR , xSy iff y=x-1

Answer 16

like its not symmetric but \( x \neq y\)

Answer 17

I would say that's antisymmetric since you could never choose \(x,y\) such that \(xSy\) and \(ySx\).

Answer 18

ohhhhh okkkk i getttt ittttttt

Answer 19

YAYYYYYYYYYYYYYYYYYY

Answer 20

THHNKKSSSS WE HAVE A FLOOOODD ON THE HOUSE SOOO ILLL BRB

Answer 21

A flood O.O? Good luck with that.