From the image, we have two sets $B$ and $C$ of real numbers defined as:

• $B = \{y \,|\, y \leq 4\}$
• $C = \{y \,|\, y > 6\}$

We are asked to find:

1. $B \cap C$ (the intersection of $B$ and $C$)
2. $B \cup C$ (the union of $B$ and $C$)

1. Intersection $B \cap C$

The intersection of two sets includes elements that belong to both sets.

• $B$ includes all real numbers $y$ such that $y \leq 4$.
• $C$ includes all real numbers $y$ such that $y > 6$.

There are no numbers that can be both $\leq 4$ and $> 6$ at the same time, so:

$B \cap C = \emptyset$

2. Union $B \cup C$

The union of two sets includes all elements that belong to either set.

• $B = (-\infty, 4]$
• $C = (6, \infty)$

The union will combine these intervals, and since there's a gap between 4 and 6, we can write:

$B \cup C = (-\infty, 4] \cup (6, \infty)$

Final Answers:

• $B \cap C = \emptyset$
• $B \cup C = (-\infty, 4] \cup (6, \infty)$

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How do you find the intersection and union of more than two sets?
2. What happens if the two sets overlap? How does that affect the intersection?
3. Can you explain the interval notation in more detail?
4. How would the problem change if $C$ was defined as $C = \{y \,|\, y \geq 6\}$?
5. What is the difference between an open interval and a closed interval?

Tip: Remember that intersections look for common elements between sets, while unions combine all elements from the sets.