Given, universal set `={x : x in N, 10 le x le 35}, A ={x in N: x le 16}` and `B={x : x gt 29}`. Find: (i) A', (ii) B'

Answer

To solve the problem, we need to find the complements of sets A and B with respect to the universal set U.

### Step-by-Step Solution:

1. **Identify the Universal Set (U)**:
The universal set is defined as:
\[ U = \{x : x \in \mathbb{N}, 10 \leq x \leq 35\} \]
This means U includes all natural numbers from 10 to 35. Therefore:
\[ U = \{10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35\} \]

2. **Identify Set A**:
Set A is defined as:
\[ A = \{x \in \mathbb{N} : x \leq 16\} \]
Therefore, A includes all natural numbers less than or equal to 16 within the universal set:
\[ A = \{10, 11, 12, 13, 14, 15, 16\} \]

3. **Identify Set B**:
Set B is defined as:
\[ B = \{x \in \mathbb{N} : x > 29\} \]
Therefore, B includes all natural numbers greater than 29 within the universal set:
\[ B = \{30, 31, 32, 33, 34, 35\} \]

4. **Find the Complement of Set A (A')**:
The complement of A, denoted as A', is defined as:
\[ A' = U - A \]
This means we take all elements in U that are not in A:
\[ A' = \{17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35\} \]

5. **Find the Complement of Set B (B')**:
The complement of B, denoted as B', is defined as:
\[ B' = U - B \]
This means we take all elements in U that are not in B:
\[ B' = \{10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29\} \]

### Final Answers:
(i) \( A' = \{17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35\} \)

(ii) \( B' = \{10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29\} \)