Let's go through the problem step-by-step.

The sets given are: $A = \langle -2, 3 \rangle \cup (5, \infty)$ $B = \langle -1, 7 \rangle$

Part (1): Set Operations on $A$ and $B$

Let's calculate each operation for the sets $A$ and $B$:

1. Union $A \cup B$: The union of $A$ and $B$ includes all elements in either $A$ or $B$.

2. Intersection $A \cap B$: The intersection includes only elements that are in both $A$ and $B$.

3. Difference $A \setminus B$: The difference $A \setminus B$ includes elements that are in $A$ but not in $B$.

4. Difference $B \setminus A$: The difference $B \setminus A$ includes elements that are in $B$ but not in $A$.

5. Complement of A $A'$: The complement of $A$ includes elements not in $A$, assuming the universal set is all real numbers.

6. Complement of B $B'$: The complement of $B$ includes elements not in $B$, assuming the universal set is all real numbers.

Part (2): Cartesian Product $A \times B$ and $B \times A$

This part requires calculating the Cartesian products of $A$ and $B$ as ordered pairs. These ordered pairs can then be plotted on a rectangular coordinate system.

I'll perform the calculations for each of these operations and display them.I've displayed the results of the set operations $A \cup B$, $A \cap B$, $A \setminus B$, and $B \setminus A$ for your review. Let me know if you would like further details on these results or any specific calculations.

Here are some related questions that might interest you:

1. How can we visualize these set operations on a number line?
2. What are the steps to calculate the complement of a set in the real number line?
3. How would the results change if $A$ or $B$ had closed or open intervals differently?
4. What is the Cartesian product of sets $A \times B$ and $B \times A$, and how would it be plotted?
5. How do these operations change when applying them to finite sets rather than intervals?

Tip: When dealing with intervals, always pay attention to whether they are open or closed, as this affects the inclusion of boundary points in the operations.