State which of the following sets are finite and which are infinite :
(i) `A={x:x inNand(x-1)(x-2)=0}`
(ii) `B={x:x inNandx^(2)=4}`
(iii) `C={x:x in N and2x-1=0}`
(iv) `E={x:x in N and x` is prime}
(v) `D={x:x inN` and x is odd}.

Answer

To determine whether each of the given sets is finite or infinite, we will analyze each set one by one.

### Step-by-Step Solution:

**(i) Set A: `A = {x : x ∈ N and (x - 1)(x - 2) = 0}`**

1. **Solve the equation**: The equation \((x - 1)(x - 2) = 0\) implies that \(x - 1 = 0\) or \(x - 2 = 0\).
- Thus, \(x = 1\) or \(x = 2\).
2. **Identify the elements of set A**: Therefore, the elements of set A are \{1, 2\}.
3. **Determine if the set is finite or infinite**: Since there are only two elements, set A is a finite set.

**Conclusion**: Set A is finite.

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**(ii) Set B: `B = {x : x ∈ N and x² = 4}`**

1. **Solve the equation**: The equation \(x^2 = 4\) gives us \(x = 2\) or \(x = -2\).
2. **Identify the elements of set B**: Since \(x\) must be a natural number (N), we only consider \(x = 2\). Thus, the elements of set B are \{2\}.
3. **Determine if the set is finite or infinite**: There is only one element, so set B is finite.

**Conclusion**: Set B is finite.

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**(iii) Set C: `C = {x : x ∈ N and 2x - 1 = 0}`**

1. **Solve the equation**: The equation \(2x - 1 = 0\) simplifies to \(2x = 1\) or \(x = \frac{1}{2}\).
2. **Identify the elements of set C**: Since \(x\) must be a natural number, and \(\frac{1}{2}\) is not a natural number, there are no elements in set C.
3. **Determine if the set is finite or infinite**: An empty set is considered finite as it has a countable number of elements (zero in this case).

**Conclusion**: Set C is finite.

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**(iv) Set E: `E = {x : x ∈ N and x is prime}`**

1. **Identify the elements of set E**: The prime numbers in natural numbers are 2, 3, 5, 7, 11, 13, 17, and so on.
2. **Determine if the set is finite or infinite**: There are infinitely many prime numbers, so set E is an infinite set.

**Conclusion**: Set E is infinite.

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**(v) Set D: `D = {x : x ∈ N and x is odd}`**

1. **Identify the elements of set D**: The odd natural numbers are 1, 3, 5, 7, 9, 11, and so on.
2. **Determine if the set is finite or infinite**: There are infinitely many odd natural numbers, so set D is an infinite set.

**Conclusion**: Set D is infinite.

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### Final Summary:
- Set A: Finite
- Set B: Finite
- Set C: Finite
- Set D: Infinite
- Set E: Infinite