7 Fill in the blanks Consider the incomplete proof below: Theorem: Let f: A→A be any function from a set to itself, and define g: A →A by g(x) = f(f(x)). Then f is injective if and only if g is injective. Proof: First, we will prove that assuming that Let x, y € A be arbitrary, and suppose g(x) = g(y), or in other words f(f(x)) = f(f(y)). By the assumption we made, this means that f(x) = f(y), and by using the same assumption again, we get x = y. Second, we will prove that suppose f(x) = f(y). Then assuming that , so g(x) = g(y); by assumption, this means that x = y. Let x, y € A be arbitrary, and Which part of the theorem is being proved in the first paragraph, and which part is being proved in the second paragraph? In the second paragraph, explain how we can go from f(x) = f(y) to g(x) = g(y).

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