Which of the following is the empty/non-empty set
`(i) {x:x"is a real number and"x^(2)-1=0}`
`{x : x"is a real number and" x^(2)+1=0}`
`(iii) {x : x "is a real number and" x^(2)-9=0}`
`(iv) {x : x"is a real number and " x^(2)=x+2}`

Answer

To determine whether each of the given sets is empty or non-empty, we will analyze each expression step by step.

### Step 1: Analyze the first set
**Set (i):** {x : x is a real number and x² - 1 = 0}

1. Solve the equation x² - 1 = 0.
- Factor the equation: (x - 1)(x + 1) = 0.
- Set each factor to zero:
- x - 1 = 0 → x = 1
- x + 1 = 0 → x = -1
2. The solutions are x = 1 and x = -1, both of which are real numbers.

**Conclusion for Set (i):** This set is non-empty because it contains the elements {1, -1}.

### Step 2: Analyze the second set
**Set (ii):** {x : x is a real number and x² + 1 = 0}

1. Solve the equation x² + 1 = 0.
- Rearranging gives x² = -1.
- The solutions are x = i and x = -i, where i is the imaginary unit.
2. Since both solutions are imaginary, they do not belong to the set of real numbers.

**Conclusion for Set (ii):** This set is empty because there are no real solutions.

### Step 3: Analyze the third set
**Set (iii):** {x : x is a real number and x² - 9 = 0}

1. Solve the equation x² - 9 = 0.
- Factor the equation: (x - 3)(x + 3) = 0.
- Set each factor to zero:
- x - 3 = 0 → x = 3
- x + 3 = 0 → x = -3
2. The solutions are x = 3 and x = -3, both of which are real numbers.

**Conclusion for Set (iii):** This set is non-empty because it contains the elements {3, -3}.

### Step 4: Analyze the fourth set
**Set (iv):** {x : x is a real number and x² = x + 2}

1. Rearranging gives x² - x - 2 = 0.
2. Factor the quadratic equation: (x - 2)(x + 1) = 0.
3. Set each factor to zero:
- x - 2 = 0 → x = 2
- x + 1 = 0 → x = -1
4. The solutions are x = 2 and x = -1, both of which are real numbers.

**Conclusion for Set (iv):** This set is non-empty because it contains the elements {2, -1}.

### Final Summary
- Set (i): Non-empty {1, -1}
- Set (ii): Empty {}
- Set (iii): Non-empty {3, -3}
- Set (iv): Non-empty {2, -1}