Let A = {` x | x in N` , x is a multiple of 2 }
B = {`x in N `, x is a multiple of 5 }
C = {`x | x in N` , x is a multiple of 10 }
The set `(A cap B) cap C ` is equal to

Options

A. A
B. `A cap C`
C. B
D. C

Answer

To solve the problem, we need to find the set \((A \cap B) \cap C\). Let's break this down step by step.

### Step 1: Define the Sets
1. **Set A**: This set contains all natural numbers that are multiples of 2.
\[ A = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, \ldots\} \]

2. **Set B**: This set contains all natural numbers that are multiples of 5.
\[ B = \{5, 10, 15, 20, 25, 30, 35, 40, 45, 50, \ldots\} \]

3. **Set C**: This set contains all natural numbers that are multiples of 10.
\[ C = \{10, 20, 30, 40, 50, 60, 70, 80, 90, 100, \ldots\} \]

### Step 2: Find \(A \cap B\)
To find \(A \cap B\), we need to identify the common elements in sets A and B.

- The multiples of 2 from set A are: \(2, 4, 6, 8, 10, 12, 14, 16, 18, 20, \ldots\)
- The multiples of 5 from set B are: \(5, 10, 15, 20, 25, 30, 35, 40, 45, 50, \ldots\)

The common elements (multiples of both 2 and 5) are:
\[ A \cap B = \{10, 20, 30, 40, 50, \ldots\} \]

### Step 3: Find \((A \cap B) \cap C\)
Now we need to find the intersection of \(A \cap B\) with set C.

- The elements of \(A \cap B\) are: \(10, 20, 30, 40, 50, \ldots\)
- The elements of set C are: \(10, 20, 30, 40, 50, 60, 70, 80, 90, 100, \ldots\)

The common elements in both sets are:
\[ (A \cap B) \cap C = \{10, 20, 30, 40, 50, \ldots\} \]

### Conclusion
The set \((A \cap B) \cap C\) is equal to the set of all natural numbers that are multiples of 10, which is set C.

Thus, we conclude:
\[ (A \cap B) \cap C = C \]

### Final Answer
\((A \cap B) \cap C = C = \{10, 20, 30, 40, 50, \ldots\}\)

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