The sets $P$ and $Q$ are defined as:

• $P = \{x \mid x \leq 4\}$
• $Q = \{x \mid x \geq 5\}$

The intersection of $P$ and $Q$, denoted by $P \cap Q$, includes elements that belong to both sets simultaneously.

Analysis:

• $P$ contains all numbers less than or equal to 4.
• $Q$ contains all numbers greater than or equal to 5.

There are no numbers that satisfy both conditions simultaneously. Therefore, the intersection is empty.

$P \cap Q = \emptyset$

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Related Questions:

1. What is the union of $P$ and $Q$?
2. How would the sets change if $P = \{x \mid x < 4\}$ and $Q = \{x \mid x > 5\}$?
3. What would happen if $P$ and $Q$ were both bounded sets instead?
4. Can the intersection of two disjoint sets ever be non-empty under certain definitions?
5. How can we represent $P$ and $Q$ graphically on a number line?

Tip: Always carefully analyze the boundary points when working with inequalities in set operations.