Question

is the following sentence true or false? cells in the breathing center monitor the amount of oxygen in the blood.

Answer 1

"breathing center?"
that is one of the most unspecific word I've ever come across so far.
Either way it's false because the amount of oxygen is determined by tissues like respiring muscles and other cells like "alveolous" which is situated in the lungs (if that's what you are calling as breathing center)

why do these cells affect the amount of oxygen?
Simple, it is due to Oxygen and Carbon dioxide Partial Pressure
and then apply the diffusion concept

Answer 2

I hope by breathing center you mean Lungs :) emm well as @KL-RC said it depends on partial pressures of Oxygen and Carbon dioxide
I don't know if it's true or false because you are unclear with 'breathing center'
By breathing center if you refer to Lungs then its False
because Lungs do not detect the amount of oxygen, with lungs you can only breath in or out thats it
The one to detect/ monitor Oxygen level in blood is basically brain
If brain detects low level of Oxygen, it'll send signals to lungs to breath more or harder
So you need to specify