To solve this problem, let's first interpret the mathematical sets and the task:

The problem defines two sets:

• $A = \{x \in \mathbb{R} \mid -3 < x \leq 1 \}$
• $B = \{ y \in \mathbb{R} \mid -2 \leq y < 3 \}$

You are asked to work with the following set operations:

1. $A \cap B$ (the intersection of sets A and B)
2. $A \cup B$ (the union of sets A and B)
3. $B - A$ (the difference between sets B and A)

Let’s compute each:

1. $A \cap B$ (Intersection of $A$ and $B$):

The intersection consists of the elements common to both sets.

• Set $A$ is the interval $(-3, 1]$.
• Set $B$ is the interval $[-2, 3)$.

So, the intersection $A \cap B$ would be the overlapping region between these two intervals:

• $A \cap B = [-2, 1]$

2. $A \cup B$ (Union of $A$ and $B$):

The union consists of all elements that are in either set.

• Union of the intervals $(-3, 1]$ and $[-2, 3)$ covers from $-3$ to $3$.

Thus, $A \cup B = (-3, 3)$.

3. $B - A$ (Difference $B$ minus $A$):

This represents the elements that are in set $B$ but not in set $A$.

• $B = [-2, 3)$
• $A = (-3, 1]$

The difference $B - A$ includes the part of $B$ that is not in $A$, which is $(1, 3)$.

Final answers:

• $A \cap B = [-2, 1]$
• $A \cup B = (-3, 3)$
• $B - A = (1, 3)$

Let me know if you'd like further clarification or details on any of these operations. Here are 5 related questions you might want to explore next:

1. What is the meaning of the symmetric difference between two sets?
2. How can we represent these intervals on the real number line?
3. What changes if one or both of the intervals are open or closed?
4. Can we express the sets $A$ and $B$ using set-builder notation differently?
5. How do we calculate the cardinality of finite sets?

Tip: In set theory, the intersection is always a subset of both sets, while the union includes all elements from both.