Which of the following are empty sets ?
(i) Set of odd natural numbers divisible by 2
(ii) `{x:xlt5andxgt7,x inN}`
(iii) `{x:x^(2)=25andx` is an odd integer}.

Answer

To determine which of the given sets are empty sets, we will analyze each set one by one.

### Step 1: Analyze the first set
**Set (i):** Set of odd natural numbers divisible by 2.

- **Definition of odd natural numbers:** Odd natural numbers are numbers like 1, 3, 5, 7, etc.
- **Divisibility by 2:** An odd number cannot be divisible by 2 because odd numbers are defined as numbers that are not divisible by 2.
- **Conclusion:** Since there are no odd natural numbers that can be divisible by 2, this set is empty.

### Step 2: Analyze the second set
**Set (ii):** {x : x < 5 and x > 7, x ∈ N}.

- **Understanding the conditions:** We need to find natural numbers (N) that are both less than 5 and greater than 7.
- **Natural numbers less than 5:** The natural numbers less than 5 are 1, 2, 3, and 4.
- **Natural numbers greater than 7:** The natural numbers greater than 7 are 8, 9, 10, etc.
- **Conclusion:** There cannot be any number that is both less than 5 and greater than 7 at the same time. Therefore, this set is also empty.

### Step 3: Analyze the third set
**Set (iii):** {x : x² = 25 and x is an odd integer}.

- **Finding values of x:** The equation x² = 25 has two solutions: x = 5 and x = -5.
- **Checking if they are odd integers:** Both 5 and -5 are odd integers.
- **Conclusion:** Since we have two odd integers (5 and -5) that satisfy the condition, this set is not empty.

### Final Conclusion:
- The empty sets among the given options are:
- Set (i): Set of odd natural numbers divisible by 2
- Set (ii): {x : x < 5 and x > 7, x ∈ N}

### Summary of Results:
- **Empty Sets:** Set (i) and Set (ii)
- **Non-empty Set:** Set (iii)

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