MAT 145 : Homework Solutions
نویسنده
چکیده
Problems from Review of Set Theory Notes 3 (c) We prove that A× (B −C) = (A×B)− (A×C). We show that A× (B −C) ⊂ (A×B)−(A×C), and (A×B)−(A×C) ⊂ A×(B−C). For the first containment, let x ∈ A× (B − C). Then x = (a, b) where a ∈ A and b ∈ B − C. We certainly have x ∈ A × B, and since the second coordinate of x is not in C, we conclude that x / ∈ A× C. For the second containment, let x ∈ (A× B)− (A× C). Then x = (a, b) where a ∈ A and b ∈ B, but x / ∈ A × C. Hence b / ∈ C. Therefore x ∈ A× (B − C).
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