I can get you started.

Part A:

There are a lot choices. Simply pick a measurement in units of customary and metric. Example: pounds vs. kilograms, miles vs. kilometers, Fahrenheit vs. Celsius, etc.

For instance, take a temperature conversion. We know that the relationship between F and C is: .

So if represented Fahrenheit and represented Celsius, then is and is 32.

Part B:

(i)

So the claim is true, as proven by algebra above.

(ii)

For , the claim is false. It is up to you to find a counterexample. Pick any arbitrary set of values for and see whether or not is equal to . Most likely not, if you picked your values strategically.

(iii)

Since the relationship between is linear (and thus bijective), the ordering of the values of are preserved during a mapping to . Thus, the smallest value of gets sent to the smallest value of . Same with the largest value and the median, etc.

(iv)

Again, find a counterexample.