Let's carefully analyze both statements.

Statement (i):

$|A \cup B| = |A - B| + |B|$ Here, $A - B$ refers to the elements in $A$ that are not in $B$. The right-hand side counts the elements in $A - B$ and all of the elements in $B$, but some elements of $A \cap B$ are being counted twice: once in $B$ and again in $A \cup B$.

Therefore, this statement is false. The elements in the intersection are overcounted.

Statement (ii):

$|A \cup B| = |A - B| + |A \cap B|$ This is the correct way to count the elements in $A \cup B$. It splits the union into:

• $A - B$ (the elements in $A$ but not in $B$),
• and $A \cap B$ (the elements common to both $A$ and $B$).

This counts each element exactly once, making this statement true.

Conclusion:

• Statement (i) is false.
• Statement (ii) is true.

Thus, the correct answer is: $\boxed{\text{Statement (i) is false; (ii) is true}}$

Would you like more details or further explanation?

Related Questions:

1. How does the inclusion-exclusion principle apply to set theory?
2. What is the formula for the cardinality of the union of three sets?
3. How can we visually represent $A \cup B$ and $A \cap B$ using a Venn diagram?
4. What is the difference between $A - B$ and $B - A$?
5. How do the properties of sets change when dealing with infinite sets?

Tip:

When working with sets, always ensure you're not double-counting elements, particularly those in the intersection.