Which of the following are the examples of empty set ?
(i) `{x:x in N,xlt7and xgt9}`
(ii) {x : x is a common point of two perpendicular lines}

(iii) {x : x is an even natural number divisible by 3}
(iv) `{x : x^(2) = 4 and x " is an odd number"}`.

Answer

To determine which of the given sets are examples of an empty set, we will analyze each option step by step.

### Step 1: Analyze the first set
**Set (i):** `{x : x in N, x < 7 and x > 9}`

- Here, we are looking for natural numbers (N) that are both less than 7 and greater than 9.
- The natural numbers less than 7 are: 1, 2, 3, 4, 5, 6.
- The natural numbers greater than 9 are: 10, 11, 12, ...
- There are no numbers that satisfy both conditions simultaneously.
- **Conclusion:** This set is empty.

### Step 2: Analyze the second set
**Set (ii):** `{x : x is a common point of two perpendicular lines}`

- Two perpendicular lines intersect at exactly one point.
- Therefore, there exists at least one common point.
- **Conclusion:** This set is non-empty.

### Step 3: Analyze the third set
**Set (iii):** `{x : x is an even natural number divisible by 3}`

- The even natural numbers are: 2, 4, 6, 8, 10, 12, ...
- Among these, the even numbers divisible by 3 are: 6, 12, 18, ...
- There are infinitely many even natural numbers that are divisible by 3.
- **Conclusion:** This set is non-empty.

### Step 4: Analyze the fourth set
**Set (iv):** `{x : x^2 = 4 and x is an odd number}`

- The equation \(x^2 = 4\) gives us the solutions \(x = 2\) and \(x = -2\).
- Both solutions are even numbers.
- Since we are looking for odd numbers, there are no solutions that satisfy both conditions.
- **Conclusion:** This set is empty.

### Final Conclusion
Based on our analysis:
- Set (i) and Set (iv) are examples of empty sets.
- Set (ii) and Set (iii) are non-empty sets.

### Summary of Results
- Empty Sets: (i) and (iv)
- Non-Empty Sets: (ii) and (iii)

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