which of the above are metrics? i think all except the 1st one is a metric since the 1st does not satisfy d(x,y) = 0 <=> x=y
if any can confirm/correct my answer possibly with justification thatd be great. also willing to pay fg to whoever can help


also need help showing that


is a metric(mainly triangle inequality)

This post was edited by 2wo1ne on Feb 22 2016 11:56pm

Correct.

Notice that the second and the third are the same :
max(x,y) - min(x,y) = x - y or y - x.
In both cases, after you square it, you get |x-y|².

For the last :
* it obviously gives a non-negative real,
* since t -> 1 / (1+t²) is a positive continuous function, the only way to have its integral being zero is when x1 = x2.
* obviously ρ(x1,x2) = ρ(x2,x1)
* for the triangle inequality, you can consider 2 cases :

if x ≤ y ≤ z (or if x ≥ y ≥ z ), you'd have ρ(x,z) = ρ(x,y) + ρ(y,z)

if y is not inbetween x and z, then :
ρ(x,z) ≤ ρ(x,y) if x < z < y or y < x < z
or, ρ(x,z) ≤ ρ(y,z) if y < x < z or z < x < y.

Thanks! Also could u help with the triangle inequality for the max/min metric. Can pay more for ur help

I think I've read wrong.

Number 2
ρ (x,y) = ( max(x,y) - min(x,y) ) ²

and number 3
ρ (x,y) = | x-y | ²

which are the same, are not metrics :

ρ (0,2) = 4
ρ (0,1) = 1
ρ (1,2) = 1

hence, ρ(0,2) ≤ ρ(0,1) + ρ(1,2) doesn't stand.
This is due to the convexity of the function x ---> x²

(actually I've read |x² - y²|, I'm sorry)

Number 4, ρ (x,y) = √ | x-y |
satisfies the triangle inequality :
for every x, y, z in R, √ | x-z | ≤ √ | x-y | + √ | y-z |

Indeed, if | x-z | is lesser than | x-y | or than | y-z |, then it's obvious.
If it is greater than both, it implies that y is inbetween x and z.
We may assume : z < y < x.

Let's square both members.
On the left : x - z.
On the right : x - y + y - z + 2*√ ( x-y ) *√ ( y-z ) = x - z + 2*√ ... *√ ...
so the right member is obviously greater than the left member.

what class is this? looks like some nice math concepts

this is my class on general topology

ah i havent taken that class...

probably wont take one for quite a while if i even do tbh... i have a feeling ill be going more the computational/applied/statistical route.

Unless you plan on going for a Ph.D. in math this is the way to go. General and advanced topology and a lot of other theory courses you take teach you some okay tricks and outside the box thinking techniques but not much is actually applicable in real life nor will it help you get a job.

Ya really kinda steers me away unfortunately because theoretical math is so cool though.