Which of the following are empty (null) sets ?
(i)`[x:xlt5andxgt7,x inN}`
(ii) `{x:x^(2)=25and` x is an odd integer}
(iii) `{x:x^(2)-2=0` and is rational ]

Answer

To determine which of the given sets are empty (null) sets, we will analyze each of the three sets provided in the question step by step.

### Step 1: Analyze the first set
**Set (i):** \[ \{ x : x < 5 \text{ and } x > 7, x \in \mathbb{N} \} \]

1. The condition states that \( x \) must be less than 5 and greater than 7 simultaneously.
2. However, no number can satisfy both conditions at the same time. Therefore, there are no natural numbers \( x \) that can fulfill this requirement.

**Conclusion for Set (i):** This set is an empty set.

### Step 2: Analyze the second set
**Set (ii):** \[ \{ x : x^2 = 25 \text{ and } x \text{ is an odd integer} \} \]

1. First, we solve the equation \( x^2 = 25 \). The solutions are \( x = 5 \) and \( x = -5 \).
2. Now, we check which of these solutions are odd integers:
- \( 5 \) is an odd integer.
- \( -5 \) is also an odd integer.
3. Since both solutions are odd integers, the set contains two elements: \( \{5, -5\} \).

**Conclusion for Set (ii):** This set is not an empty set.

### Step 3: Analyze the third set
**Set (iii):** \[ \{ x : x^2 - 2 = 0 \text{ and } x \text{ is rational} \} \]

1. First, we solve the equation \( x^2 - 2 = 0 \). The solutions are \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
2. Next, we check if these solutions are rational numbers:
- \( \sqrt{2} \) is an irrational number.
- \( -\sqrt{2} \) is also an irrational number.
3. Since neither solution is rational, there are no elements that satisfy the condition.

**Conclusion for Set (iii):** This set is an empty set.

### Final Summary
- Set (i) is an empty set.
- Set (ii) is not an empty set.
- Set (iii) is an empty set.