<h2>Right answer: I and II only </h2>
If we already have the formula to find how temperature measured in degrees Fahrenheit, relates to a temperature in degres Celsius:
    (1)
We can know the formula to find how temperature measured in degrees Celsius, relates to a temperature in degres Fahrenheit, only by isolating F:
    (2)
Having both formulas, let’s begin:
I) If we want to prove that a temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of 5/9 degree Celsius:
Beginning with 
:

This means: 
Now we are going to increase 1 degree Farenheit. In other words, we are going to use 
:

This means: 
Calculating the difference between 
   and  
:
>>>>This is equal to 5/9 degree Celsius, hence is correct
II) If we want to prove that a temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit:
Beginning with 
:
  
  
This means: 
  
Now we are going to increase 1 degree Celsius. In other words, we are going to use 
:
  
  
This means: 
  
Calculating the difference between 
   and  
:
>>>>This is a proof of the statement, hence is also correct.
<h2 />
III) If we want to prove that a temperature increase of 5/9 degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius:
Beginning with 
:
  
  
This means: 
  
Now we are going to add 
:

And use this value in the Celsius formula:
 
This means: 
In other words: An increase in 5/9 degree Fahrenheit is equivalent to a temperature increase of 1.30 degree Celsius, <u>not 1 degree Celsius</u>.
Therefore this statement is incorrect.