Final answer:
To prove that the function ϕ is one-to-one, we need to show that for every pair of elements € and €', if ϕ(€) = ϕ(€'), then € = €'. This can be done by assuming that ϕ(€) = ϕ(€') and deriving a contradiction. By showing that the only way for the equation to hold is if € = €', we prove that ϕ is one-to-one.
Step-by-step explanation:
In order to prove that the function ϕ is one-to-one, we need to show that for every pair of elements € and €', if ϕ(€) = ϕ(€'), then € = €'.
To do this, let's assume that ϕ(€) = ϕ(€'). Using the definition of ϕ given in the question, this means that €^€' = €'^€. Taking the logarithm of both sides, we get log€^€' = log€'^€. Using the property of logarithms, we can rewrite this as €'log€ = €log€'. Since € and €' are relatively prime to , the only way for this equation to hold is if € = €', thus proving that ϕ is one-to-one.