Let A={1,2,3,4} B= {3,4,5,6},C={5,6,7,8} and D={7,8,9,10}.
(i) `AuuB`
(ii) `AuuC`
(iii) `BuuC`
(iv) `BuuD`
(v) `AuuBuuC`
(vi) `AuuBuuD`
(vii) `BuuCuuD`.

Answer

To solve the problem step by step, we will find the union of the given sets as specified in the question.

### Given Sets:
- \( A = \{1, 2, 3, 4\} \)
- \( B = \{3, 4, 5, 6\} \)
- \( C = \{5, 6, 7, 8\} \)
- \( D = \{7, 8, 9, 10\} \)

### Step-by-Step Solution:

**(i) \( A \cup B \)**
To find \( A \cup B \), we combine all elements from both sets, ensuring that we do not repeat any elements.
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( B \): \( 3, 4, 5, 6 \)

Combining these, we have:
\[ A \cup B = \{1, 2, 3, 4, 5, 6\} \]

**(ii) \( A \cup C \)**
Now, we find \( A \cup C \).
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( C \): \( 5, 6, 7, 8 \)

Combining these, we have:
\[ A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8\} \]

**(iii) \( B \cup C \)**
Next, we calculate \( B \cup C \).
- Elements in \( B \): \( 3, 4, 5, 6 \)
- Elements in \( C \): \( 5, 6, 7, 8 \)

Combining these, we have:
\[ B \cup C = \{3, 4, 5, 6, 7, 8\} \]

**(iv) \( B \cup D \)**
Now, we find \( B \cup D \).
- Elements in \( B \): \( 3, 4, 5, 6 \)
- Elements in \( D \): \( 7, 8, 9, 10 \)

Combining these, we have:
\[ B \cup D = \{3, 4, 5, 6, 7, 8, 9, 10\} \]

**(v) \( A \cup B \cup C \)**
Next, we calculate \( A \cup B \cup C \).
- From \( A \): \( 1, 2, 3, 4 \)
- From \( B \): \( 3, 4, 5, 6 \)
- From \( C \): \( 5, 6, 7, 8 \)

Combining these, we have:
\[ A \cup B \cup C = \{1, 2, 3, 4, 5, 6, 7, 8\} \]

**(vi) \( A \cup B \cup D \)**
Now, we find \( A \cup B \cup D \).
- From \( A \): \( 1, 2, 3, 4 \)
- From \( B \): \( 3, 4, 5, 6 \)
- From \( D \): \( 7, 8, 9, 10 \)

Combining these, we have:
\[ A \cup B \cup D = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]

**(vii) \( B \cup C \cup D \)**
Finally, we calculate \( B \cup C \cup D \).
- From \( B \): \( 3, 4, 5, 6 \)
- From \( C \): \( 5, 6, 7, 8 \)
- From \( D \): \( 7, 8, 9, 10 \)

Combining these, we have:
\[ B \cup C \cup D = \{3, 4, 5, 6, 7, 8, 9, 10\} \]

### Final Answers:
1. \( A \cup B = \{1, 2, 3, 4, 5, 6\} \)
2. \( A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8\} \)
3. \( B \cup C = \{3, 4, 5, 6, 7, 8\} \)
4. \( B \cup D = \{3, 4, 5, 6, 7, 8, 9, 10\} \)
5. \( A \cup B \cup C = \{1, 2, 3, 4, 5, 6, 7, 8\} \)
6. \( A \cup B \cup D = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
7. \( B \cup C \cup D = \{3, 4, 5, 6, 7, 8, 9, 10\} \)