Given sets `A={1,3,5,7,9}, B={0,2,4,6}` and `C={7,8,9}`. Which of the following may be taken as universal set for all the three sets A,B and C?

Options

A. A. {0,1,2,3,4,5,6,7,8}
B. B. {1,2,3,4,5,6,7,8,9}
C. C. {1,2,3,4,5,6,7,8,9,10}
D. D. {0,1,2,3,4,5,6,7,8,9,10}

Answer

To determine a universal set for the given sets A, B, and C, we need to find the union of these sets. The universal set should contain all the elements from sets A, B, and C.

### Step-by-Step Solution:

1. **Identify the Given Sets:**
- Set A = {1, 3, 5, 7, 9}
- Set B = {0, 2, 4, 6}
- Set C = {7, 8, 9}

2. **Find the Union of the Sets:**
- The union of sets A, B, and C is represented as A ∪ B ∪ C. This union includes all unique elements from each set.
- Start by listing the elements from each set:
- From Set A: 1, 3, 5, 7, 9
- From Set B: 0, 2, 4, 6
- From Set C: 7, 8, 9 (Note: 7 and 9 are already included in Set A)

3. **Combine All Unique Elements:**
- Combine all the unique elements from the three sets:
- Start with Set A: {1, 3, 5, 7, 9}
- Add elements from Set B: {0, 2, 4, 6}
- Add elements from Set C: {8} (7 and 9 are already included)
- The combined set will be: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

4. **Determine the Universal Set:**
- The universal set U can be defined as U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
- Since the universal set can also include additional elements, we can also consider including 10, making it U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

5. **Evaluate the Options:**
- Option A: {0, 1, 2, 3, 4, 5, 6, 7, 8} - Missing 9, so this is not a universal set.
- Option B: {1, 2, 3, 4, 5, 6, 7, 8, 9} - Missing 0, so this is not a universal set.
- Option C: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - Missing 0, so this is not a universal set.
- Option D: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - Contains all elements from A, B, and C, plus an extra element (10), making it a valid universal set.

### Conclusion:
The correct option for the universal set that includes all elements from sets A, B, and C is Option D: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.