Find the (a) union (b) intersection of each of the following pairs of sets:
(i) `A={x:x inZ^(+)andx^(2)gt7},B={1,2,3}`
(ii) `A={x:x inZ^(+)},B= x inZandx lt0}`
(iii) `A={x:x inNand4 ltx lt9}`.

Answer

Let's solve the problem step by step for each pair of sets.

### (i) Given Sets:
- \( A = \{ x : x \in \mathbb{Z}^+ \text{ and } x^2 > 7 \} \)
- \( B = \{ 1, 2, 3 \} \)

**Step 1: Determine the elements of set A.**
- \( x^2 > 7 \) implies \( x > \sqrt{7} \).
- Since \( \sqrt{7} \approx 2.645 \), the smallest integer \( x \) can take is 3.
- Therefore, \( A = \{ 3, 4, 5, 6, \ldots \} \) (all positive integers greater than or equal to 3).

**Step 2: Identify the elements of set B.**
- \( B = \{ 1, 2, 3 \} \).

**Step 3: Find the intersection \( A \cap B \).**
- The common elements between \( A \) and \( B \) are \( \{ 3 \} \).
- Thus, \( A \cap B = \{ 3 \} \).

**Step 4: Find the union \( A \cup B \).**
- The union combines all distinct elements from both sets.
- \( A \cup B = \{ 1, 2, 3, 4, 5, 6, \ldots \} \).
- This can be expressed as \( A \cup B = \{ x : x \in \mathbb{Z}^+ \} \) (all positive integers).

### (ii) Given Sets:
- \( A = \{ x : x \in \mathbb{Z}^+ \} \)
- \( B = \{ x : x \in \mathbb{Z} \text{ and } x < 0 \} \)

**Step 1: Identify the elements of set A.**
- \( A = \{ 1, 2, 3, 4, 5, \ldots \} \) (all positive integers).

**Step 2: Identify the elements of set B.**
- \( B = \{ \ldots, -3, -2, -1 \} \) (all negative integers).

**Step 3: Find the intersection \( A \cap B \).**
- There are no common elements between positive integers and negative integers.
- Thus, \( A \cap B = \emptyset \) (the empty set).

**Step 4: Find the union \( A \cup B \).**
- The union combines all distinct elements from both sets.
- \( A \cup B = \{ \ldots, -3, -2, -1, 1, 2, 3, 4, \ldots \} \).
- This can be expressed as \( A \cup B = \{ x : x \in \mathbb{Z} \text{ and } x \neq 0 \} \) (all integers except zero).

### (iii) Given Sets:
- \( A = \{ x : x \in \mathbb{N} \text{ and } 4 < x < 9 \} \)
- \( B = \{ x : x \in \mathbb{N} \} \)

**Step 1: Identify the elements of set A.**
- The natural numbers between 4 and 9 are \( 5, 6, 7, 8 \).
- Thus, \( A = \{ 5, 6, 7, 8 \} \).

**Step 2: Identify the elements of set B.**
- \( B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, \ldots \} \) (all natural numbers).

**Step 3: Find the intersection \( A \cap B \).**
- The common elements between \( A \) and \( B \) are \( \{ 5, 6, 7, 8 \} \).
- Thus, \( A \cap B = \{ 5, 6, 7, 8 \} \).

**Step 4: Find the union \( A \cup B \).**
- The union combines all distinct elements from both sets.
- \( A \cup B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, \ldots \} \).
- This can be expressed as \( A \cup B = \{ x : x \in \mathbb{N} \} \) (all natural numbers).

### Summary of Results:
1. (i) \( A \cap B = \{ 3 \} \), \( A \cup B = \{ x : x \in \mathbb{Z}^+ \} \)
2. (ii) \( A \cap B = \emptyset \), \( A \cup B = \{ x : x \in \mathbb{Z} \text{ and } x \neq 0 \} \)
3. (iii) \( A \cap B = \{ 5, 6, 7, 8 \} \), \( A \cup B = \{ x : x \in \mathbb{N} \} \)