To find $|P \cup Q|$, which represents the number of unique elements in the union of sets $P$ and $Q$, we’ll go through the following steps:

Step 1: Identify the Elements in Each Set

We have:

• $P = \{ f, i, x, t, u, r, e \}$
• $Q = \{ r, e, l, a, x \}$

Step 2: Form the Union of $P$ and $Q$

The union, $P \cup Q$, includes all unique elements from both $P$ and $Q$.

So, $P \cup Q = \{ f, i, x, t, u, r, e, l, a \}$

Step 3: Count the Elements in $P \cup Q$

Counting each unique element:

• There are 9 elements: $f, i, x, t, u, r, e, l, a$.

Conclusion

$|P \cup Q| = 9$

Would you like a breakdown of why duplicates are excluded or more on set operations?

Here are five related questions that expand on this concept:

1. How would you find the intersection of sets $P$ and $Q$?
2. If another set $S = \{m, n, o\}$ is introduced, how would $|P \cup Q \cup S|$ be calculated?
3. What is the formula for calculating $|P \cup Q|$ using $|P|$, $|Q|$, and $|P \cap Q|$?
4. How would you represent $P \setminus Q$, and what does it signify?
5. How does the concept of complement relate to set union and intersection?

Tip: In set theory, $|P \cup Q|$ counts only unique elements from both sets, effectively ignoring duplicates.