Final answer:
To prove that if the composition of two functions is an injection, the two original functions must also be injections, we assume the composition is an injection and show that each original function must be an injection as well.
Step-by-step explanation:
Proving: If the composition of two functions is an injection, then the two original functions must be injections too.
To prove this statement, we need to show that if the composition of two functions is an injection, then each of the original functions must be injections as well.
Let's assume that the composition of two functions f and g, denoted as (f ∘ g), is an injection. This means that for any two input values x and y, if (f ∘ g)(x) = (f ∘ g)(y), then x = y.
Now, let's consider the two original functions, f and g. If f(x) = f(y), then we can substitute these values into the composition equation (f ∘ g)(x) = (f ∘ g)(y).
Since the composition is an injection, it follows that x = y. Therefore, the original function f must also be an injection.
Similarly, we can apply the same logic to the function g and show that it must also be an injection.