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Question
Answers

If A={x: x is a multiple of 2}, B={x: x is a multiple of 5} and C={x:x is a multiple of 10}, then \[A \cap (B \cap C)\] is equal to
A. A
B. B
C. C
D. {x: x is a multiple of 100}

Answer Verified Verified
Hint: Here the sets are given in builder form, we have to convert it in the roster form. Then we have to find the common elements in sets B and C first and then the common elements in set A & set $B \cap C$.

Complete step-by-step answer:

A is a set of multiple of 2
$ \Rightarrow $ A= {2, 4, 6, 8, 10, 12, 14, 16…}
B is a set of multiple of 5
$ \Rightarrow $ B= {5, 10, 15, 20, 25, 30, 35…}
C is a set of multiple of 10
$ \Rightarrow $ C= {10, 20, 30, 40, 50, 60, 70…}
Now,
$ \Rightarrow \left( {B \cap C} \right) = $ {10, 20, 30, 40…} = $C$
$ \Rightarrow A \cap \left( {B \cap C} \right) = A \cap C = $ {10, 20, 30, 40…} = $C$
So option (C) $C$ is correct.

Note: In these types of questions, convert the sets in set builder form to roster form to make the solution easier, then find the quantity required in the order mentioned in the question. The final solution set can be represented in either roster or set builder form.