Let's solve this step by step based on the information provided in the image.

Given Sets:

• $A = \langle -2, 3 \rangle \cup (5, \infty)$
• $B = \langle -1, 7 \rangle$

where:

• $\langle -2, 3 \rangle$ represents a closed interval on $A$,
• $(5, \infty)$ represents an open interval on $A$,
• $\langle -1, 7 \rangle$ represents a closed interval on $B$.

Problem 1: Set Operations

For the operations in Part (1), let's proceed with each set operation.

1. $A \cup B$: Union of $A$ and $B$.

   • $A = \langle -2, 3 \rangle \cup (5, \infty)$
   • $B = \langle -1, 7 \rangle$
   • The union will combine all intervals where $A$ or $B$ have values.

2. $A \cap B$: Intersection of $A$ and $B$.

   • We will find the overlapping regions of $A$ and $B$.

3. $A \setminus B$: Elements in $A$ but not in $B$.

   • This involves excluding the elements of $B$ from $A$.

4. $B \setminus A$: Elements in $B$ but not in $A$.

   • This excludes elements in $A$ from $B$.

5. $A \setminus A$: This is essentially the empty set since $A \setminus A = \emptyset$.

6. $B'$: Complement of $B$.

   • This includes all elements outside of $B$.

Problem 2: Cartesian Products

For the operations in Part (2), we find Cartesian products:

1. $A \times B$: Cartesian product of $A$ and $B$.

   • This will involve pairs $(a, b)$ where $a \in A$ and $b \in B$.

2. $B \times A$: Cartesian product of $B$ and $A$.

   • This will involve pairs $(b, a)$ where $b \in B$ and $a \in A$.

Solution Steps:

1. Calculate each set operation for Part 1.
2. Display results on a number line for each operation.
3. Calculate Cartesian products for Part 2.
4. Plot Cartesian products in a rectangular coordinate system.

Would you like me to proceed with detailed calculations and visual representations of each step?

Here are some additional questions you might be interested in based on this problem:

1. How do you calculate the intersection of two intervals?
2. What is the significance of open vs. closed intervals in set operations?
3. How can we represent Cartesian products graphically?
4. How would we find the complement of a set in a real number line context?
5. What are some practical applications of Cartesian products in set theory?

Tip: Always remember to carefully distinguish between open and closed intervals when performing set operations, as they affect the elements included in each interval.