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A,B,C are non empty sets then the value ...

A,B,C are non empty sets then the value of `(A cap B)cap (Bcap C)cap (Ccap A)` is

A

`A cup B cup C`

B

`A cup (B cup C)`

C

`A cap B cap C`

D

`A cap (B cup C)`

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The correct Answer is:
To solve the problem of finding the value of \((A \cap B) \cap (B \cap C) \cap (C \cap A)\), we will follow these steps: ### Step 1: Understand the Problem We need to find the intersection of three sets, which are themselves intersections of pairs of sets. The expression can be rewritten as: \[ (A \cap B) \cap (B \cap C) \cap (C \cap A) \] ### Step 2: Break Down the Intersections We can break down the expression into parts: 1. \(A \cap B\) - This is the set of elements that are in both set A and set B. 2. \(B \cap C\) - This is the set of elements that are in both set B and set C. 3. \(C \cap A\) - This is the set of elements that are in both set C and set A. ### Step 3: Visualize with a Venn Diagram To visualize the intersections, we can draw a Venn diagram with three overlapping circles representing sets A, B, and C. The areas where the circles overlap represent the intersections of the sets. ### Step 4: Find the Common Intersection Now, we need to find the intersection of all three pairs: - The intersection of \(A \cap B\) with \(B \cap C\) will give us the elements that are common in all three sets A, B, and C. - The intersection of \(C \cap A\) will further refine this to only those elements that are present in all three sets. ### Step 5: Conclusion The final result of \((A \cap B) \cap (B \cap C) \cap (C \cap A)\) is simply: \[ A \cap B \cap C \] This is because the only elements that are common to all three intersections are those that are present in all three sets A, B, and C. ### Final Answer Thus, the value of \((A \cap B) \cap (B \cap C) \cap (C \cap A)\) is: \[ A \cap B \cap C \]
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