Problem 4.5 (Grade a “Proof”). Study the following claim as well as the “proof”: Claim. For all sets A, B, C and D, (A ×

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Problem 4.5 (Grade a “Proof”). Study the following claim as wellas the “proof”: Claim. For all sets A, B, C and D, (A × B) − (C ×D) = (A − C) × (B − D). “Proof”. Let A, B, C and D be sets. Let (x,y) ∈ (A × B) − (C × D), so that (x, y) ∈ A × B and (x, y) ∈/ C × D.Thus x ∈ A, x /∈ C, y ∈ B and y /∈ D. In other words, x ∈ A − C andy ∈ B − D; hence (x, y) ∈ (A − C) × (B − D). Since every element in(A × B) − (C × D) is in (A − C) × (B − D), we conclude (A × B) − (C× D) = (A − C) × (B − D). This completes the proof. ✷ Complete thefollowing questions concerning the above claim and “proof”: (1)Determine whether the “proof ” is rigorous. Identify the issues inthe “proof”, if any. (2) Determine whether the claim is true orfalse. Justify the answer in part (3). (3) If the the claim is trueand the “proof ” is not rigorous, then provide a correct andrigorous proof. If the claim is false, give a concretecounterexample.