If D={x:x is divisible by 2 and 3 and `0 lt x lt 20}` and B={x:x is a multiple of 6 and `0 lt x lt 25}` then D-B is

Options

A. `phi`
B. {2,3}
C. {6,12}
D. {6,12,18}

Answer

To solve the problem, we need to find the sets D and B based on the given conditions and then compute the difference D - B.

### Step 1: Define the Set D
Set D is defined as:
\[ D = \{ x : x \text{ is divisible by 2 and 3 and } 0 < x < 20 \} \]

To find the elements of D, we need to identify numbers between 1 and 19 that are divisible by both 2 and 3. A number is divisible by both 2 and 3 if it is divisible by their least common multiple (LCM), which is 6.

The multiples of 6 that are less than 20 are:
- 6 (6 × 1)
- 12 (6 × 2)
- 18 (6 × 3)

Thus, the set D is:
\[ D = \{ 6, 12, 18 \} \]

### Step 2: Define the Set B
Set B is defined as:
\[ B = \{ x : x \text{ is a multiple of 6 and } 0 < x < 25 \} \]

The multiples of 6 that are less than 25 are:
- 6 (6 × 1)
- 12 (6 × 2)
- 18 (6 × 3)
- 24 (6 × 4)

Thus, the set B is:
\[ B = \{ 6, 12, 18, 24 \} \]

### Step 3: Compute D - B
Now we need to find the difference D - B, which includes all elements of D that are not in B.

Since:
- \( D = \{ 6, 12, 18 \} \)
- \( B = \{ 6, 12, 18, 24 \} \)

We see that all elements of D are also present in B. Therefore, when we subtract B from D, we have:
\[ D - B = \{ \} \]
This means D - B is the empty set.

### Final Answer
Thus, the result of D - B is:
\[ D - B = \emptyset \]

---