1. Suppose f: A → B is an injective function. Prove that f-¹(f(C)) = C for all CCA. 2. Suppose A, B are sets, f : A → B is a function, and D₁, D₂ C B. In lectures we noted that the inverse image is well behaved with respect to set operations: f-¹ (D₁ D₂) = f¯¹ (D₁) f¯¹ (D₂), f-¹ (D₁U D₂) = f¯¹ (D₁) Uf-¹ (D₂), and f¯¹ (D₁ A D₂) = f¯¹ (D₁) A f¯¹ (D₂). The image is not so well behaved. Yet one of the above three identities continues to hold with f replacing f-1 (and with D₁, D₂ replaced by subsets of A). Which is it? Can you prove this identity?

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