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 which of the given sets are empty or non-empty, we will analyze each set one by one.

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

To find the elements of this set, we solve the equation:
\[ x² - 1 = 0 \]
Factoring gives:
\[ (x - 1)(x + 1) = 0 \]
Setting each factor to zero gives:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

Thus, the set is:
\[ \{1, -1\} \]
This is a non-empty set.

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

To find the elements of this set, we solve the equation:
\[ x² + 1 = 0 \]
Rearranging gives:
\[ x² = -1 \]

Since there are no real numbers whose square is negative, this equation has no solutions. Therefore, the set is:
\[ \emptyset \]
This is an empty set.

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

To find the elements of this set, we solve the equation:
\[ x² - 9 = 0 \]
Factoring gives:
\[ (x - 3)(x + 3) = 0 \]
Setting each factor to zero gives:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

Thus, the set is:
\[ \{3, -3\} \]
This is a non-empty set.

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

To find the elements of this set, we rearrange the equation:
\[ x² - x - 2 = 0 \]
Factoring gives:
\[ (x - 2)(x + 1) = 0 \]
Setting each factor to zero gives:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

Thus, the set is:
\[ \{2, -1\} \]
This is a non-empty set.

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

### Final Answer
- Empty Set: (ii) {x : x is a real number and x² + 1 = 0}
- Non-Empty Sets: (i), (iii), (iv)