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For all sets A,B and C : Is (A-B) cap(C-...

For all sets A,B and C : Is (A-B) `cap(C-B)=(AcapC)-B`?
Justify your answer :

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To determine whether \((A - B) \cap (C - B) = (A \cap C) - B\) for all sets \(A\), \(B\), and \(C\), we can use set operations and Venn diagrams. Here’s a step-by-step solution: ### Step 1: Understand the Left-Hand Side We start with the left-hand side of the equation: \((A - B) \cap (C - B)\). - **Definition of \(A - B\)**: This represents the elements that are in \(A\) but not in \(B\). - **Definition of \(C - B\)**: This represents the elements that are in \(C\) but not in \(B\). ### Step 2: Visualize with Venn Diagram Draw a Venn diagram with three overlapping circles representing sets \(A\), \(B\), and \(C\). - Shade the area representing \(A - B\) (the part of \(A\) that does not overlap with \(B\)). - Shade the area representing \(C - B\) (the part of \(C\) that does not overlap with \(B\)). - The intersection \((A - B) \cap (C - B)\) will be the area that is shaded in both regions. ### Step 3: Understand the Right-Hand Side Now consider the right-hand side of the equation: \((A \cap C) - B\). - **Definition of \(A \cap C\)**: This represents the elements that are common to both \(A\) and \(C\). - **Subtracting \(B\)**: We then remove any elements that are also in \(B\) from this intersection. ### Step 4: Visualize Right-Hand Side with Venn Diagram Using the same Venn diagram, shade the area representing \(A \cap C\) (the overlap between \(A\) and \(C\)). - Then, remove the shaded area that overlaps with \(B\). This gives us \((A \cap C) - B\). ### Step 5: Compare the Two Shaded Areas Now, compare the shaded regions from both sides: - The shaded area from \((A - B) \cap (C - B)\) represents elements that are in \(A\) and \(C\) but not in \(B\). - The shaded area from \((A \cap C) - B\) also represents elements that are in both \(A\) and \(C\) but not in \(B\). ### Conclusion Since both shaded areas represent the same set of elements, we can conclude that: \[ (A - B) \cap (C - B) = (A \cap C) - B \] Thus, the statement is justified.
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