Let N be the universal set . Write the complements of the following sets :
(i) `{x:x inN and`x is an odd}
(ii) `{x:x inN andx` is even}
(iii) `{x:x` is a prime number}
(iv) `{x:x inNand x=3n` for some `n inN}`
(v) `{x:x inNandx` is a perfect square}
(vi) `{x:x inNandx` is a perfect cube}
(vii) `{x:x inNandx+5=7}`
(viii) `{x:x inNand2x+5=111}`
(ix) `{x:x inNandxge6}`
(x) `{x:x inNandx` is divisible by 3 and 5}.

Answer

To find the complements of the given sets in the universal set \( N \) (the set of natural numbers), we will follow the definition of the complement of a set. The complement of a set \( A \) is defined as all elements in the universal set that are not in \( A \).

Let's denote the universal set \( N \) as the set of all natural numbers:
\[ N = \{ 1, 2, 3, 4, 5, 6, \ldots \} \]

Now, we will find the complements for each of the given sets:

### (i) Set: \( \{ x : x \in N \text{ and } x \text{ is odd} \} \)
- The odd numbers in \( N \) are \( \{ 1, 3, 5, 7, \ldots \} \).
- The complement will be the even numbers:
\[ \{ x : x \in N \text{ and } x \text{ is even} \} \]

### (ii) Set: \( \{ x : x \in N \text{ and } x \text{ is even} \} \)
- The even numbers in \( N \) are \( \{ 2, 4, 6, 8, \ldots \} \).
- The complement will be the odd numbers:
\[ \{ x : x \in N \text{ and } x \text{ is odd} \} \]

### (iii) Set: \( \{ x : x \in N \text{ and } x \text{ is a prime number} \} \)
- The prime numbers in \( N \) are \( \{ 2, 3, 5, 7, 11, \ldots \} \).
- The complement will be the composite numbers (and 1, which is neither prime nor composite):
\[ \{ x : x \in N \text{ and } x \text{ is composite or } x = 1 \} \]

### (iv) Set: \( \{ x : x \in N \text{ and } x = 3n \text{ for some } n \in N \} \)
- This set contains multiples of 3: \( \{ 3, 6, 9, 12, \ldots \} \).
- The complement will be all natural numbers that are not multiples of 3:
\[ \{ x : x \in N \text{ and } x \neq 3n \text{ for any } n \in N \} \]

### (v) Set: \( \{ x : x \in N \text{ and } x \text{ is a perfect square} \} \)
- The perfect squares in \( N \) are \( \{ 1, 4, 9, 16, 25, \ldots \} \).
- The complement will be all natural numbers that are not perfect squares:
\[ \{ x : x \in N \text{ and } x \text{ is not a perfect square} \} \]

### (vi) Set: \( \{ x : x \in N \text{ and } x \text{ is a perfect cube} \} \)
- The perfect cubes in \( N \) are \( \{ 1, 8, 27, 64, \ldots \} \).
- The complement will be all natural numbers that are not perfect cubes:
\[ \{ x : x \in N \text{ and } x \text{ is not a perfect cube} \} \]

### (vii) Set: \( \{ x : x \in N \text{ and } x + 5 = 7 \} \)
- Solving \( x + 5 = 7 \) gives \( x = 2 \).
- The complement will be all natural numbers except 2:
\[ \{ x : x \in N \text{ and } x \neq 2 \} \]

### (viii) Set: \( \{ x : x \in N \text{ and } 2x + 5 = 111 \} \)
- Solving \( 2x + 5 = 111 \) gives \( 2x = 106 \) or \( x = 53 \).
- The complement will be all natural numbers except 53:
\[ \{ x : x \in N \text{ and } x \neq 53 \} \]

### (ix) Set: \( \{ x : x \in N \text{ and } x \geq 6 \} \)
- This set includes \( \{ 6, 7, 8, 9, \ldots \} \).
- The complement will be all natural numbers less than 6:
\[ \{ x : x \in N \text{ and } x < 6 \} \]

### (x) Set: \( \{ x : x \in N \text{ and } x \text{ is divisible by 3 and 5} \} \)
- A number divisible by both 3 and 5 is also divisible by 15: \( \{ 15, 30, 45, \ldots \} \).
- The complement will be all natural numbers that are not divisible by 15:
\[ \{ x : x \in N \text{ and } x \neq 15n \text{ for any } n \in N \} \]

### Summary of Complements
1. \( \{ x : x \in N \text{ and } x \text{ is even} \} \)
2. \( \{ x : x \in N \text{ and } x \text{ is odd} \} \)
3. \( \{ x : x \in N \text{ and } x \text{ is composite or } x = 1 \} \)
4. \( \{ x : x \in N \text{ and } x \neq 3n \text{ for any } n \in N \} \)
5. \( \{ x : x \in N \text{ and } x \text{ is not a perfect square} \} \)
6. \( \{ x : x \in N \text{ and } x \text{ is not a perfect cube} \} \)
7. \( \{ x : x \in N \text{ and } x \neq 2 \} \)
8. \( \{ x : x \in N \text{ and } x \neq 53 \} \)
9. \( \{ x : x \in N \text{ and } x < 6 \} \)
10. \( \{ x : x \in N \text{ and } x \neq 15n \text{ for any } n \in N \} \)